PHYSICS OF FLUIDS 22, 055106 共2010兲

Large-eddy simulation of turbulent collision of heavy particles in isotropic turbulence Guodong Jin,1 Guo-Wei He,1,a兲 and Lian-Ping Wang1,2 1

State Key Laboratory for Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 2 Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716-3140, USA

共Received 1 October 2009; accepted 30 March 2010; published online 20 May 2010兲 The small-scale motions relevant to the collision of heavy particles represent a general challenge to the conventional large-eddy simulation 共LES兲 of turbulent particle-laden flows. As a first step toward addressing this challenge, we examine the capability of the LES method with an eddy viscosity subgrid scale 共SGS兲 model to predict the collision-related statistics such as the particle radial distribution function at contact, the radial relative velocity at contact, and the collision rate for a wide range of particle Stokes numbers. Data from direct numerical simulation 共DNS兲 are used as a benchmark to evaluate the LES using both a priori and a posteriori tests. It is shown that, without the SGS motions, LES cannot accurately predict the particle-pair statistics for heavy particles with small and intermediate Stokes numbers, and a large relative error in collision rate 共up to 60%兲 may arise when the particle Stokes number is near StK = 0.5. The errors from the filtering operation and the SGS model are evaluated separately using the filtered-DNS 共FDNS兲 and LES flow fields. The errors increase with the filter width and have nonmonotonic variations with the particle Stokes numbers. It is concluded that the error due to filtering dominates the overall error in LES for most particle Stokes numbers. It is found that the overall collision rate can be reasonably predicted by both FDNS and LES for StK ⬎ 3. Our analysis suggests that, for StK ⬍ 3, a particle SGS model must include the effects of SGS motions on the turbulent collision of heavy particles. The spectral analysis of the concentration fields of the particles with different Stokes numbers further demonstrates the important effects of the small-scale motions on the preferential concentration of the particles with small Stokes numbers. © 2010 American Institute of Physics. 关doi:10.1063/1.3425627兴 I. INTRODUCTION

Particle-laden turbulent flows are encountered in a wide range of engineering and environmental problems;1,2 for example, pollutant or aerosol particle dispersion, warm rain droplet formation in the atmosphere, and fluidization and turbulent mixing in combustion processes.3,4 Understanding the hydrodynamics of such flows is the basis for many applications in engineering design, environmental protection, and weather forecasting. Particularly, turbulence-mediated collision of heavy particles is vital to efficiency and conversion rates in these processes.5 Large-eddy simulation 共LES兲 has emerged as a promising tool for simulating turbulent flows in general and, in recent years, has also been applied to particle-laden turbulence with some successes.6 The motion of heavy particles is much more complicated than fluid elements, and therefore, LES of turbulent flow laden with heavy particles encounters new challenges. In LES, only large-scale eddies are explicitly resolved and the effects of unresolved, small-scale or subgrid scale 共SGS兲 eddies on the large-scale eddies are parametrized. The SGS turbulent velocity field is not available. a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected] and [email protected] Telephone: 86-1082543969. Fax: 86-10-82543408.

1070-6631/2010/22共5兲/055106/13/$30.00

The effects of SGS turbulent velocity fields on particle motions have been studied in channel flows or isotropic turbulent flows by Yeh and Lei,7 Wang and Squires,8 Armenio, Piomelli, and Fiorotto,9 Yamamoto et al.,10 Kuerten,11 Shotorban and Mashayek,12,13 Fede and Simonin,14 Berrouk et al.,15 Bini and Jones,16 Pozorski and Apte,17 among others. The Eulerian deconvolution method11,12 or Lagrangian stochastic approach13,15,17 is usually used to model the effects of SGS motions on particle dispersion. The missing of SGS turbulent velocity does not pose a serious problem for quantifying the properties that are mainly governed by large-scale turbulent eddies, i.e., single-particle statistics such as oneparticle dispersion coefficient.7–9 However, the turbulent collision of heavy particles is a small-scale process, where the small-scale velocity field plays an important role. The collision-related relative motions and pair distribution of heavy particles with the Stokes numbers 共StK ⬅ p / K, i.e., the ratio of heavy particle response time to flow Kolmogorov time兲 on the order of 1 is mainly determined by the smallscale eddies.18–20 These particle-pair statistics and thus collision rate of heavy particles are unlikely to be properly simulated in LES. This is one of the main challenges in LES of particle-laden turbulent flows. Another issue is that the flow field in LES is more coherent in space and more correlated in time than either the fully resolved flow field or direct nu-

22, 055106-1

© 2010 American Institute of Physics

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Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang

merical simulation 共DNS兲 one due to the filtering operation and overdissipation of the commonly used eddy viscosity SGS model.21–23 The flow field from the LES with an eddy viscosity SGS model may not recover the properties especially needed for collision-rated statistics of heavy particles. Fede and Simonin14 studied the effects of SGS turbulent motions on particle-pair statistics of nonsettling, monodispersed heavy particles in an isotropic turbulent flow field. DNS was used to generate a turbulent flow at Taylor microscale Reynolds number Re = 34.1. The filtered-DNS 共simply denoted as FDNS hereafter兲 flow field was obtained from the DNS one by filtering the small-scale fluid motions at different cutoff scales. One-way coupling was assumed and particles were treated as perfect-elastic hard spheres. They focused on the effects of SGS motions on both one- and twoparticle statistics of heavy particles. They concluded that the effects of SGS motions must be included when the SGS Stokes number p / ␦ [email protected] is less than 5.0, where ␦ [email protected] is the SGS Lagrangian integral time scale seen by heavy particles. A very interesting result is that the effect of filtering leads to a reduction in preferential concentration if p ⬍ 0.5␦ [email protected], but an increase in the preferential concentration if p ⬎ 0.5␦ [email protected] For the case of p ⬍ 0.5␦ [email protected], the eddies at the dissipation scales play an important role in particle clustering. Filtering reduces the intensity of the dissipation-scale eddies and thus reduces the level of preferential concentration. For the case of 5␦ [email protected] ⬎ p ⬎ 0.5␦ [email protected], the particles respond to the eddies ranging from the dissipation-scale eddies to the larger-scale ones. By removing the dissipationscale eddies, the larger-scale eddies may actually become more coherent in space and better correlated in time, although the maximum vorticity intensity is reduced. This implies that the actual level of preferential concentration is governed not only by the vorticity intensity but also by the vorticity distribution. Fede et al.24 then developed a onepoint stochastic Lagrangian model to recover the SGS motions seen by heavy particles. The model predicts a good match to DNS data for the particle kinetic energy. The collision-related statistics has not been quantitatively compared and assessed simultaneously using DNS, FDNS, and LES in previous studies. The aim of this paper is to investigate the effects of the spectral eddy viscosity SGS model and the SGS fluid motions on particle-pair statistics such as radial distribution function at contact, radial relative velocity at contact, and collision rate. By understanding these effects, we hope to develop some insights into the dynamic interaction of heavy particles with SGS flow fields, as well as a guidance on a better particle SGS model aimed at improving particle-pair statistics in LES. This paper is organized as follows. An overview of the governing equations and simulation methods is given in Sec. II The numerical results about the effects of SGS motions on collision-related statistics of heavy particles are presented in Sec. III. Conclusions on LES of turbulent collision of heavy particles can be found in Sec. IV.

II. GOVERNING EQUATIONS AND SIMULATION METHODS

The equations governing fluid flow and motion of heavy particles are described in this section, along with our numerical methods. A. DNS method

The Navier–Stokes equations for isotropic turbulence are

冉

冊

u p 1 = u ⫻ − ⵜ + u2 + ⵜ2u + f共x,t兲, t 2

共1兲

ⵜ · u = 0,

共2兲

where u denotes the velocity, = ⵜ ⫻ u is the vorticity, p is the pressure, is the fluid density, and is the fluid kinematical viscosity. The flow is driven and maintained by a random forcing f共x , t兲, which is nonzero only at low wavenumbers in Fourier space 兩k兩 ⬍ 冑8.25,26 The DNS of isotropic turbulence is performed using a standard pseudospectral method on N3 grids covering a periodic box of side L = 2. Here, N = 256. In Fourier space, Eqs. 共1兲 and 共2兲 can be represented as 共k ⱕ kmax兲,

冉

冊

+ k2 uˆ 共k,t兲 = P共k兲F共u ⫻ 兲 + ˆf共k,t兲, t

共3兲

where uˆ 共k , t兲 is the Fourier coefficient or the fluid velocity in Fourier space, F denotes a Fourier transformation, the projection tensor P = 兵P jm其 = ␦ jm − k jkm / k2 共j , m = 1 , 2 , 3兲 projects F共u ⫻ 兲 onto the plane normal to k and eliminates the pressure term in Eq. 共1兲 with the aid of the continuity equation 共2兲, and kmax is the maximum wavenumber. The wavenumber components in Fourier space are k j = n j共2 / L兲, where n j = −N / 2 , . . . , −1 , 0 , 1 , . . . , N / 2 − 1 for j = 1 , 2 , 3. Aliasing errors are removed using the two-thirds truncation method. The spatial resolution is monitored by the value of kmax, where is the Kolmogorov length scale. The value of kmax is typically larger than 1.1 in the present simulation. The Fourier coefficients are advanced in time using a second-order Adams–Bashforth method for the nonlinear term and an exact integration for the linear viscous term. The time step is chosen to ensure that the Courant– Friedrichs–Lewy number is less than 0.5 for numerical stability and accuracy.26,27 B. FDNS velocity fields

The FDNS velocity field is obtained from the DNS velocity field by truncating the Fourier coefficients larger than the cutoff wavenumber kcf, kcf

˜u共x,t兲 =

兺

uˆ 共k,t兲eik·x ,

共4兲

兩k兩=k0

where ˜u共x , t兲 is the filtered velocity in physical space and kcf is the cutoff wavenumber in FDNS. k0 = 1 is the lowest wavenumber. FDNS can be regarded as an ideal LES to study the

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Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision

effects of SGS eddies on the statistics of particle motions since FDNS does not contain any eddy viscosity SGS model errors.14 C. LES method

The LES of isotropic turbulence is performed on the coarser grids using the same pseudospectral method and large-scale forcing scheme as DNS. The governing equation in LES is given by 共兩k兩 ⱕ kc, where kc is the maximum wavenumber in LES兲

再

冎

+ 关 + e共k兩kc兲兴k2 uˆ共k,t兲 = P共k兲F共u ⫻ 兲 + ˆf共k,t兲, t

TABLE I. The parameters in DNS and LES. DNS 共2563兲

LES 共643兲

Reynolds number rms fluid velocity u⬘ 共Effective兲 dissipation rate Minimum length scale

102.05 19.34

¯ 18.52

3771.4 0.0135

3434.8 0.0238

Minimum time scale

0.0037

0.0055

Minimum velocity scale

3.6272

4.327

Eulerian integral time scale Molecular viscosity

0.050 0.0488

0.056 0.0488

Method

共5兲 where u and are the resolved velocity and vorticity in physical space, respectively. The term e共k 兩 kc兲k2uˆ共k , t兲 on the left-hand side models the net effects of SGS motions on the resolved motions. The spectral eddy viscosity SGS model21,22 is used in this paper, given by

冑

E共kc兲 , kc

共6兲

with

+e 共k/kc兲 = CK−3/2关0.441 + 15.2 exp共− 3.03kc/k兲兴.

共7兲

The spectral viscosity e共k 兩 kc兲 depends on the wavenumber k, the maximum wavenumber kc as well as E共kc兲, the value of the energy spectrum function at kc. Here, E共kc兲 in Eq. 共6兲 is dynamically evaluated from the LES fluid field. CK = 2.5 is taken in this paper. D. Motion of inertial particles

The dispersed phase is composed of N p spherical heavy particles with a diameter smaller than the Kolmogorov length scale , d p = 0.5. In general, there are several different forces acting on a heavy particle suspended in a nonuniform and unsteady turbulent flow field.28 When the ratio of particle density to fluid density is much larger than 1 共 p / f Ⰷ 1兲, the equation of motion for a heavy particle can be approximated as26 dx p共t兲 = v p共t兲, dt

共8兲

dv p共t兲 u关x p共t兲,t兴 − v p共t兲 + w0 , = dt p

共9兲

III. NUMERICAL RESULTS A. Statistics of DNS and LES flow fields

Table I lists Eulerian statistics of the DNS and LES flow fields in this study. The root mean square 共rms兲 of turbulent fluctuation velocity u⬘ and the average dissipation rate are computed from the three dimensional turbulent energy spectrum function shown in Fig. 1, u⬘ =

冑

1 具uiui典 = 3

冑冕 2 3

kmax

共10兲

E共k兲dk,

k0

2

10

where x p共t兲 and v p共t兲 are the particle position and velocity at time t, w0 is the particle terminal or settling velocity under gravity in otherwise quiescent fluid, w0 = g p, p is the particle Stokes response time, and g is the gravitational acceleration. The particle terminal velocity is set to w0 = 共−vK , 0 , 0兲 in this study, where vK is the Kolmogorov velocity obtained from DNS. The linear Stokes drag force is assumed due to the low particle Reynolds number, u关x p共t兲 , t兴 is the fluid velocity seen by a heavy particle which is obtained from the DNS, FDNS, and LES flow fields, respec-

101 E(k)

e共k兩kc兲 = +e 共k/kc兲

tively, by a six-point Lagrangian interpolation scheme in each direction.27 The equation of motion 关Eq. 共9兲兴 is integrated with a fourth-order Adams–Bashforth method for the particle velocity and then a fourth-order Adams–Moulton method for the particle location in Eq. 共8兲. The particles are assumed to have identical size and density, and each moves independent of others 共i.e., the so-called ghost particle model兲. This is appropriate for turbulent geometric collision.29,30 All particle pairs and collision search are conducted using the efficient cell-index method and the concept of linked lists.18,31

100

DNS 2563 LES 643 k-5/3

-1

10

10-2 0 10

1

10 k

2

10

FIG. 1. The energy spectra of the simulated flows in 2563 DNS and 643 LES.

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055106-4

=

Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang

冕

kmax

2k2E共k兲dk.

共11兲

k0

In LES, the molecular viscosity coefficient in Eq. 共11兲 is replaced by the effective viscosity + ve共k 兩 kc兲 to calculate the effective dissipation rate, e =

冕

kc

2关 + ve共k兩kc兲兴k2E共k兲dk,

共12兲

k0

which represents the sum of the molecular dissipation rate of large-scale modes and the dissipation rate of the SGS eddies. The dissipation rate in LES 共643兲 listed in Table I is the effective one. The actual or molecular dissipation rate of the large-scale motion is 1626.6. As the spectral eddy viscosity coefficient ve共k 兩 kc兲 varies with the wavenumber, an average SGS viscosity coefficient can be estimated using the enstrophy spectrum 2k2E共k兲 as weight factors, ¯SGS =

兰kkc 2ve共k/kc兲k2E共k兲dk 0

兰kkc 2k2E共k兲dk

共13兲

.

0

This average eddy viscosity coefficient in LES, ¯SGS = 0.0543, is only slightly larger than the molecular viscosity coefficient used in DNS, = 0.0488. This could be due to the moderate flow Reynolds number realized in DNS, given that the same large-scale flow forcing and flow domain size are applied in LES and DNS. The smallest 共Kolmogorov兲 length, time, and velocity scales in DNS are

= 共3/兲0.25,

K = 共/兲0.5,

vK = 共兲0.25 .

共14兲

In order to obtain the minimum characteristic scales in LES flow field, the effective viscosity 共 + ¯SGS兲 and effective dissipation rate e are used to replace the molecular viscosity and molecular dissipation rate , respectively, in Eq. 共14兲 to represent the smallest characteristic length, time, and velocity scales as

e =

冋

共 + ¯SGS兲3 e

册

0.25

,

K,e =

冋

共 + ¯SGS兲 e

册

0.5

,

vK,e = 关e共 + ¯SGS兲兴0.25 .

共15兲

Ideally, e = if there was no SGS model error in LES. Figure 1 plots the resulting energy spectra of DNS and LES flow fields in log-log coordinates. It is shown that the energy spectrum of LES flow field decays faster than that of DNS flow field at high wavenumbers. This is due to the overdissipation in the spectral eddy viscosity SGS model. In this study, the cutoff wavenumber in FDNS is taken as kcf = 21, corresponding to the LES at the grid resolution of 643. B. Preferential concentration of heavy particles

It is well known that, unlike passive tracers, heavy particles are distributed nonuniformly in a turbulent flow field due to their interaction with vortical and straining motions of local flows. Maxey32 developed an asymptotic analysis of the preferential concentration of particles with weak inertia. If

we consider the particle distribution from an Eulerian viewpoint and the particle velocity field is then defined as v共x , t兲, its divergence field is ⵜ · v = − p

冉

冊

u j ui 1 = − p s2i,j − 2 , 2 xi x j

共16兲

where = 兩ⵜ ⫻ u兩 and sij = 21 共ui / x j + u j / xi兲 are the local fluid vorticity and strain rate tensor, respectively. Equation 共16兲 indicates that particle velocity field is compressible even in an incompressible fluid velocity field and particles accumulate in regions of low vorticity and high strain rate. Figure 2 shows the preferential concentration of heavy particles at different particle Stokes numbers in a DNS flow field. N p = 1.2⫻ 106 heavy particles are randomly released into a statistically stationary flow field. The spatial distribution of particles evolves with time and particle preferential concentration develops in the flow field. After about four times of the Eulerian integral time scale, the accumulation is balanced by randomly turbulent mixing and the preferential concentration reaches its quasiequilibrium state. The particle terminal velocity is w0 = 共−vK , 0 , 0兲 for all kinds of particles, where vK is the Kolmogorov velocity. The background shows the contour of local flow vorticity at z = obtained from DNS 共2563 , Re = 102.05兲. The locations of all particles in a slice with a thickness of 2 / 256 centered at z = are projected onto the x-y plane. Three Stokes numbers are considered: StK = 0.1, 1.0, 10.0. The degree of nonuniformity depends on the particle Stokes numbers. For the Stokes number StK = 0.1, the heavy particle response time is much shorter than the Kolmogorov time scale; therefore, particles respond quickly to the changes of the flow field and nearly follow the fluid motion. In this case, large-scale fluid motion serves to efficiently disperse the particles. However, even at StK = 0.1, the inertial bias is noticeable and more particles are found in the regions of low flow vorticity 关see the dark regions in Fig. 2共a兲兴. For StK = 1, particle motion is strongly affected by small-scale eddies and particles are centrifuged out of the vortical structures, leading to a strong nonuniformity 关see Fig. 2共b兲兴. As heavy particle response time increases, particles respond to the eddies with larger time scales relative to the Kolmogorov eddies. The level of accumulation starts to drop when StK ⬎ 1. At StK = 10, particles accumulate into large-scale structures and are mixed by small eddies. Therefore, particles return back to a more uniform distribution 关see Fig. 2共c兲兴. Since the particles have a fixed terminal velocity of vK in the negative x direction, patches of particles aligned in the vertical direction are visible in Figs. 2共b兲 and 2共c兲 as a result of the preferential sweeping.26 In general, the particle concentration field contains more large-scale structures as the Stokes number increases. This is demonstrated in Fig. 3, which plots the energy spectra of concentration fluctuations at the Stokes numbers StK = 0.1, 0.5, 1.0, 3.0, 5.0, and 10.0, ranging from small to large inertial particles. The particle concentration C共x , t兲 is defined as the number of particles in a local cell of center x. In the present calculations, 1283 coarse grids are used to smooth the particle concentration field, although 2563 fine grids are used

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055106-5

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision ω 6 5 4 3 2 1

6 5

5 4

g x

x

4 3

2

1

1 1

2

3 y

4

5

g

3

2

0 0

ω 6 5 4 3 2 1

6

0 0

6

1

(a) StK= 0.1

2

4

5

6

(b) StK = 1.0 ω 6 5 4 3 2 1

6 5 4 x

3 y

g

3 2 1 0

0

1

2

3 y

4

5

6

(c) StK = 10.0 FIG. 2. 共Color兲 The preferential concentration of particles with different Stokes numbers in a slice centered at z = with a thickness of 2 / 256. The background is the vorticity contour at z = of DNS flow field, which is normalized by the average vorticity. The arrow below the legend represents the direction of gravity and the particle terminal velocity is set to w0 = 共−vK , 0 , 0兲, where vK is the Kolmogorov velocity. For a small Stokes number in 共a兲, the particles almost follow the trajectories of fluid particles and uniformly distribute in the flow field. For a large Stokes number in 共c兲, the particles are too heavy to respond to the eddies and also uniformly distribute in the flow field. For an intermediate Stokes number in 共b兲, the preferential concentration is most obvious.

for the flow field in DNS. The difference between the particle concentration C共x , t兲 at time t and the initial concentration C共x , 0兲 is defined as the concentration fluctuations, which largely remove the statistical noises in practical computations. Since the probability distribution function 共PDF兲 of the initial concentration C共x , 0兲 is set to be a Poisson distribution, the difference C共x , t兲 − C共x , 0兲 measures the fluctuations of concentration C共x , t兲 relative to the uniform concentration C共x , 0兲. When the particle concentration field becomes statistically stationary, the concentration fluctuations are trans-

formed into the Fourier space and the squared magnitudes of the modes in each wavenumber shell within the two radii between k − 0.5 and k + 0.5 are summed to yield the concentration energy spectrum Ec共k兲, where k = 1 , 2 , 3 , . . .. Ten time frames with a time interval equal to 0.2TE are used to obtain an average energy spectrum for each Stokes number, where TE is the Eulerian integral time. The area under each curve is equal to 具C2典 − 具C2共0兲典 = 兰⬁1 Ec共k兲dk, where 具C2共0兲典 denotes the concentration variance of heavy particles at initial time. It measures the degree of nonuniformity of the particle con-

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055106-6

6 5

-2

6 5

4

Ec(k)

10

3 2

-4

1

10-5

(a)

5

6

3

2 4 5

2

6

1 2 3 4 5 6

100

Stk=0.1 Stk=0.5 Stk=1.0 Stk=3.0 Stk=5.0 Stk=10.0

4 3

0.01

1

6

4 3

0.005

6

3 3

102

(b)

FIG. 3. 共Color online兲 Wavenumber spectra of particle concentration fields of heavy particles with different Stokes numbers. 共a兲 Logarithmic scales; 共b兲 linear scales.

3

2 4 2 4

5

2

0

Stk=0.1 Stk=0.5 Stk=1.0 Stk=3.0 Stk=5.0 Stk=10.0

3 5

6 2

2 1

101 k

1 2 3 4 5 6

4

5

1

1

10

3 24 6

2 -3

0.015

4 5 6 3 5 3 6 2 2

Ec(k)

10

Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang

6 1

5

20

40

60

k

centration for a given Stokes number. This area increases with Stk, reaches a maximum at Stk = O共1兲, and then decreases with Stk. It is observed from Fig. 3共a兲 that as the Stokes numbers increase, the peaks of concentration spectra move from larger wavenumbers to smaller ones with more concentration energy contained at small wavenumbers. The peak wavenumber gives an indication of the length scale at which these eddies contribute most significantly to the concentration field. Figure 3共b兲 shows that the small scales contain more concentration energy than the large ones for StK ⱕ 3, while the small scales contain less concentration energy than the large ones for StK ⬎ 3. Therefore, Fig. 3 implies that the small-scale eddies are important to the particle concentration for StK ⱕ 3, and simply neglecting small-scale eddies in the conventional LES may lead to incorrect predictions. A particle SGS model to account for the effects of fluid motions at small scales on particle motions is needed in LES. In order to study the effects of average particle number concentration on the energy spectrum of particle concentration field, we doubled the number of particles to N p = 2.4 ⫻ 106 while keeping the same mesh resolution, i.e., 1283 when defining the local number concentration. The energy spectra of nondimensional particle concentration fields, Eˆc共k兲, are plotted in Fig. 4, where the nondimensional particle concentration field is defined as C共x , t兲 / 具C典 and 具C典 = N p / 共2兲3 is the volume-averaged particle concentration. In Fig. 4, the thin solid lines with symbols are the results obtained with N p = 1.2⫻ 106 and the thick dashed lines are the

results obtained with N p = 2.4⫻ 106. We observe that the wavenumber spectra of the nondimensional particle concentration are not sensitive to N p for all Stokes numbers considered, although the curves obtained with the larger N p are smoother as expected. C. Radial distribution function

The nonuniformity of particle concentration shown in Fig. 2 has an important consequence on turbulent collision rates, which can be quantified in terms of the radial distribution function.18,33 For monodispersed particle system in a statistically isotropic flow field, the radial distribution function g共r兲 is defined as the ratio of the number density of particle pairs with interparticle distance from r to r + dr, Npair共r兲 / Vs, to the average number density of particle pairs in the flow domain, g共r兲 =

Npair共r兲 Vs

冒

1 2 N p共N p

共2兲

− 1兲 3

共17兲

,

where Vs = 34 关共r + dr兲3 − r3兴 is the volume of the spherical shell, Npair共r兲 is the number of particles in the shell, and N p共N p − 1兲 / 2 is the total possible particle pairs for N p monodispersed particles.18 For a uniformly distributed system, the radial distribution function approaches 1. Figure 5 shows the transient variation of the radial distribution function of particles with StK = 1.0 in the 2563 DNS flow field. At beginning, the particles are uniformly distributed so that g共r / R兲 = 1, where R is

K=10.0 0.04 StSt K=5.0

StK=3.0

101

0.03 0.02

g(r/R)

Êc(k)

StK=1.0

10

StK=0.5

0.01

0

t=0.0 t=0.025 t=0.05 t=0.1 t=0.2 t=0.25

StK=0.1

0 0

20

40

60

k FIG. 4. 共Color online兲 Wavenumber spectra of nondimensional particle concentration. The thin solid lines with symbols are the results obtained with N p = 1.2⫻ 106 and the thick dashed lines are the results with N p = 2.4⫻ 106. The wavenumber spectra for the two particle numbers almost collapse for each Stokes number considered.

10

-1

1

3 r/R

5

7

9

FIG. 5. Variation in the radial distribution function g共r / R兲 with the separation distance at different times in flow field of 2563 DNS, where the particle Stokes number StK = 1.0. g共r / R兲 exhibits a power-law scaling with r.

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055106-7

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision

15

result from the random large-scale forcing. The time averaged value of g共R兲 does not depend on the used particle number. The results reported in the following are based on 4 ⫻ 105 particles and are averaged over time after the quasiequilibrium state is obtained.

g(R)

10

D. Effects of filter width on the collision-related statistics

5

The collision rate of monodispersed particles is formulated as18

0

0

2

4

t/TE

6

8

10

FIG. 6. Variation in the radial distribution function at contact g共R兲 with time in flow field of 2563 DNS and the particle Stokes number StK = 1.0. The flow reaches a statistically steady state with fluctuations after about four times of the Eulerian integral time scale, TE = 0.05. The fluctuation is related to the finite numbers of particles, which is shown in Fig. 7.

the geometric collision radius, R = d p for monodispersed particles, and d p is the particle diameter. As time increases, particle concentration deviates from the initially uniform distribution and g共r / R兲 increases monotonically with time. Finally, the concentration field approaches a statistically stationary state after about four times of the Eulerian integral time scale TE. At the quasiequilibrium, we found that g共r / R兲 follows a power-law scaling with r / R, consistent with previous results.29 It is also observed in Fig. 5 that g共r / R兲 still follows the power-law scaling even in the case of sedimentation. For geometric collision rate, the radial distribution function at contact, i.e., g共r / R = 1兲, is of interest.33 Figure 6 shows the variation in g共r / R = 1兲 with time. At long times, g共r / R兲 fluctuates with time due to the finite number of particles and turbulent mixing. The magnitude of fluctuation decreases as the particle number increases. Figure 7 shows that the magnitude of fluctuation for 106 particles is smaller than that of 4 ⫻ 105 particles in the 643 LES flow field where the particle Stokes number StK = 1.3. Some of the fluctuations

10

g(R)

8 6

5

4x10 particles 1x106 particles

4 2 0 0

2

4

6

8

t/TE FIG. 7. The effect of particle numbers on the magnitude of fluctuation of g共R兲 in the flow of 643 LES, StK = 1.3. The fluctuation magnitude of g共R兲 for 106 particles is much smaller than that for 4 ⫻ 105. The Eulerian integral time scale of LES flow field TE = 0.056.

2

n 具N˙c典 = 2R2具兩wr共R兲兩典具g共R兲典 0 , 2

共18兲

where n0 = N p / 共2兲3 is the average particle number density, N p is the total particle number in the flow domain, 具兩wr共R兲兩典 is the averaged radial relative velocity of collision pairs at contact, 具兩wr共R兲兩典 =

冓冏 冏冔 w共R兲 · r r

,

共19兲

where r is the vector connecting the centers of the collision pair, r = 兩r兩, and w共R兲 = v p1 − v p2 is the relative velocity between the two particles. The angle brackets denote an ensemble average over all collision pairs. In order to study the effects of different filter widths on particle collision-related statistics, we calculate 具g共R兲典, 具兩wr共R兲兩典, and 具N˙c典 of particles with different Stokes numbers StK = 0.5, 1, and 5 in different FDNS flow fields with the cutoff wavenumbers kcf = 64, kcf = 42, and kcf = 21, respectively. The statistics of the energy, enstrophy, and energy dissipation rate in 2563 DNS, FDNS, and 643 LES flow fields are given in Table II. As the energy spectrum of FDNS for k ⱕ kcf is the same as that of the DNS, typically very little turbulent kinetic energy is contained within high wavenumbers 共k ⬎ kcf兲. The percentage of energy filtered out in the FDNS is negligible except that the cutoff wavenumber is sufficiently low. For the largest filter width at kcf = 21 in the present study, only 3% of the kinetic energy is filtered out, so the rms turbulent velocity in the FDNS flow field differs very little from that of the DNS flow field. However, as enstrophy is defined as 兰kkcf2k2E共k兲dk, a large portion of enstrophy is 0 distributed at high wavenumbers, thus a large percentage of enstrophy is filtered out in FDNS. For the case of kcf = 21, about 45.7% of enstrophy is filtered out. With the same spatial resolution in the FDNS 共kcf = 21兲 and 643 LES, the total turbulent energy of LES is lower than that of FDNS, and the enstrophy is even lower due to the overdissipation of the eddy viscosity SGS model. The relative errors in 具g共R兲典, 具兩wr共R兲兩典, and 具N˙c典, using DNS results as a benchmark, are shown in Fig. 8. For a given Stokes number, the relative errors of these statistics increase with increasing filter width, and the relative errors for 具g共R兲典, 具兩wr共R兲兩典, and 具N˙c典 are not the same. For example, when kcf = 21 and StK = 0.5, the relative errors for 具g共R兲典, 具兩wr共R兲兩典, and 具N˙c典 are ⫺0.37, ⫺0.29, and ⫺0.57, respectively. For a given filter width, the filter operation has different effects on the motion of particles with

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055106-8

Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang TABLE II. Kinetic energy, enstrophy in DNS 共2563兲, FDNS, and LES 共643兲. Method

DNS 共2563兲

kcf = 64

kcf = 42

kcf = 21

kcf

1.15

0.864

0.567

0.285

¯

561.3 77 290.6

561.0 74 409.8

559.5 66 833.7

544.2 41 994.6

514.5 33 330.6

Actual dissipation rate

3771.4

3631

3261

2049

1626.6

Energy filtered out 共%兲

¯

0.05

0.32

3.04

8.3

Enstrophy filtered out 共%兲

¯

3.72

13.5

45.7

56.9

Energy Enstrophy

different Stokes numbers. The reason is that particles with different inertia are related to different length scales of eddies.34 It implies that filtering operation can result in a higher or lower prediction of 具g共R兲典 for different Stokes ˙ 典. It numbers and always lower prediction of 具兩w 共R兲兩典 and 具N r

c

0.1

0.1

0

0

-0.1

StK=0.5

-0.2

StK=1.0 StK=5.0

rErr of

rErr of

is important to note that, for small Stokes numbers, the relative error on the relative velocity is much larger than the relative amount of turbulent kinetic energy removed, since the main contribution to the particle relative motion comes from small-scale fluid motion. Our simulation results about the effects of filtering on particle collision rate are consistent with those of Fede and Simonin.14 We have included the data in Fig. 28 of Fede and Simonin14 in Fig. 8共c兲 for comparison. Some of the quantitative difference between their data and ours could be due to the fact that Fede and Simonin14 assumed zero settling and the Stokes numbers are not the same. The effects of inertia on particle-pair statistics will be discussed in detail later.

-0.3 -0.4 0.2 (a)

E. Effects of eddy viscosity SGS model on collision-related statistics

A spectral eddy viscosity SGS model 关see Eqs. 共6兲 and 共7兲兴 is used to close the filtered Navier–Stokes equations in LES. From Fig. 1, one can see that the spectral eddy viscosity SGS model overdissipates the turbulent kinetic energy, especially near the cutoff wavenumbers, making the vorticity field in LES more coherent than that in FDNS with the same spatial resolution. As particle turbulent collision rate is related to the particle preferential concentration and the averaged radial relative velocity, we will first study the influences of SGS motions and the spectral eddy viscosity SGS model on the two aforementioned quantities and then the collision rates using both a priori and a posteriori tests, i.e., we will compare the results from the DNS and the ones from the FDNS and the LES flow fields, respectively. This study will be carried out over a wide range of the Stokes numbers.

-0.1

StK=0.5

-0.2

StK=1.0 StK=5.0

-0.3 0.4

LES 共643兲

0.6 0.8 kcf/kmax

-0.4 0.2

1

0.4

(b)

0.2

rErr of

0 .

StK=1.0 StK=5.0 StK=0.37 StK=0.88 StK=4.73

-0.4

(c)

1

StK=0.5

-0.2

-0.6 0.2

0.6 0.8 kcf/kmax

FIG. 8. Effects of filter width on the collision-related statistics of different Stokes numbers, where rErr denotes the relative error of the result from FDNS to that from DNS, rErr= 共␣FDNS − ␣DNS兲 / ␣DNS, and ␣ represents a statistical quantity and the open symbols in 共c兲 are our numerical results and the solid symbols are obtained from Fig. 28 of Fede and Simonin 共Ref. 14兲. 共a兲 The radial distribution function at contact 具g共R兲典; 共b兲 the radial relative velocity at contact 具兩wr共R兲兩典; 共c兲 the ˙ 典. particle collision rate 具N c

0.4

0.6 0.8 kcfη

1

1.2

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055106-9

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision

13 0.4

DNS FDNS

9

rErr of

LES

5

FDNS LES

0.2

FIG. 9. Effects of filtering operation and spectral eddy viscosity model on 具g共R兲典 of different Stokes numbers in LES 共643兲, FDNS 共kcf = 21兲, and DNS 共2563兲 flow fields. 共a兲 具g共R兲典 variation with Stk; 共b兲 relative error of 具g共R兲典 from LES and FDNS to the DNS results.

0

-0.2 -0.4

1 10-2

(a)

10-1

100 St K

101

10-2

102

(b)

10-1

100 StK

1. Radial distribution function

A nonuniform distribution of particles 共the accumulation effect兲 at the Stokes numbers near one is mainly induced by small-scale eddies which are directly related to the cutoff wavenumber in FDNS and related to both the cutoff wavenumber and the spectral eddy viscosity SGS model in LES. For kcf = 21, the space resolution in 2563 DNS is four times of the ones in FDNS and LES 共⌬xLES = ⌬xFDNS = 4⌬xDNS兲. Figure 9共a兲 shows the dependence of 具g共R兲典 from the DNS, FDNS, and LES on the Stokes numbers, and Fig. 9共b兲 plots the relative errors 共rErr兲 of the FDNS and LES to the DNS results. Figure 9 shows that there are three regimes for 具g共R兲典. In the limit cases of both StK → 0 and StK → ⬁, particles tend to be uniformly distributed and 具g共R兲典 → 1 since particles with a very small Stokes number are distributed like fluid elements and particles with a very large Stokes number are not responsive to flow structures. For the intermediate Stokes numbers, particles interact with a range of fluid eddies and tend to accumulate in the regions of low vorticities, leading to the large values of 具g共R兲典 with a peak value around StK = 1 in the DNS. This trend is consistent with the previous observations.18,33 Our results from Figs. 8共a兲 and 9 are qualitatively similar to Fig. 22 of Fede and Simonin,14 where a different measure of preferential concentration was used. For very small Stokes numbers, StK ⬍ 1, the effect of filtering is to decrease the particle accumulation as small-scale eddies most relevant to accumulation have been removed. A very interesting observation from Fig. 9 is that the magnitude of the relative error in 具g共R兲典 for the FDNS flow has its maximum near StK = 0.5, with a maximum reduction of as much as 40%. For StK ⬎ 0.5, the magnitude of the relative error decreases with increasing the Stokes number, indicating that the SGS eddies make less and less contribution to the preferential concentration. For StK ⬎ 1, the FDNS yields a slightly higher 具g共R兲典 than the DNS result, indicating that the accumulation of these particles is more controlled by resolved eddies and the SGS eddies may act to randomize the particle distribution. Therefore, filtering out the random small eddies increases 具g共R兲典. For StK ⬎ 10, the particles are not responsive to the SGS eddies so the FDNS yields accurate results. Compared with the FDNS at the same spatial resolution, the LES predicts an even lower value of 具g共R兲典 for small Stokes numbers StK ⬍ 1. This is due to the overdissipation of the SGS model which tends to damp the eddies near the

101

102

SGS, making them contribute less to preferential concentration. Interestingly, for the intermediate Stokes numbers 1 ⬍ StK ⬍ 10, a higher 具g共R兲典 is realized when compared to the FDNS. This results from the fact that the resolved large-scale eddies are more coherent in space and time, particles have a longer interaction time with low vorticity regions, leading to a somewhat stronger preferential concentration. For the particles with very large Stokes numbers, StK ⬎ 10, the results from the DNS, FDNS, and LES are similar. The peak values of 具g共R兲典 in the FDNS and LES occur at a slightly larger Stokes numbers than that in the DNS. This is expected as the effective time scales for strongest vortical structures in the FDNS and LES are larger than K. 2. The radial relative velocity

For particles with large Stokes numbers, the velocities of the collision pair are uncorrelated and the relative velocity at contact is mainly determined by the energy-contained eddies.18 The energy-contained eddies are supposed to be at large scales and well resolved by LES. However, in the limit when particle inertia goes to zero, the relative velocity of particles is essentially determined by the local gradient of fluid velocity field provided that the collision radius R is small. In this limit, Saffman and Turner35 provided a prediction for 具兩wr共R兲兩典 in terms of the turbulent kinetic energy dissipation rate or the enstrophy of the flow field. As shown in Table II, although the relative percentage of turbulent kinetic energy that is filtered out in LES is negligible, a significant amount of enstrophy has been filtered out. Thus, we expect that a large error could arise in the LES prediction of the relative velocities for particles at very small Stokes numbers. Figure 10共a兲 shows the dependence of 具兩wr共R兲兩典 on the Stokes number, and Fig. 10共b兲 plots the relative errors of the FDNS and LES to the DNS results. It can be seen that 具兩wr共R兲兩典 is small when the particle Stokes number is small 共StK ⬍ 1兲. The reason for this behavior is that, in this limit, the two particles are strongly correlated as they are transported by large eddies, and only small-scale eddies are effective in generating particle relative motion. It is also observed that the quantity 具兩wr共R兲兩典 increases rapidly with the particle Stokes number for StK ⬍ 20. As the particle Stokes number increases, the large-scale turbulent fluctuations become effective in generating relative motion as different eddies encountered by the particles during their passage to a geometric

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055106-10

Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang

101 DNS

0

FDNS

B

ST, FDNS

C

ST, LES

0.5

A

A

A

B C

B C

B C

0.4A

A

0.3 B C

B C

0.2

B C

0.1 0

A

0.1

B C

0.2

0.3

0.4

LES ST, FDNS

B

-0.2

ST, LES

C B

B

B -0.15

C

-0.2

C

-0.25

-0.4

B

B

B

B

-0.35 C

C

C

C

-0.3

-0.4

0.5

StK

10

-2

10

-1

(a)

10 StK

0

10

1

10

-0.6

2

10

-2

10

-1

10 StK

(b)

0

contact can all contribute to the relative motion 共i.e., the memory effect兲. For StK ⬎ 20, the relative velocity begins to decrease slowly with the Stokes number, as there are no more large-scale fluctuations in the flow to disturb the particle motion.18 One can observe that the relative velocity in the FDNS is smaller than that in the DNS, but larger than that in the LES. Therefore, the relative error in the LES is larger than that in the FDNS for particles with small and intermediate Stokes numbers. As particle inertia increases, the radial relative velocity is affected more and more by the large-scale turbulence. The relative errors for FDNS and LES decrease with increasing particle Stokes number. We shall now analyze the various behaviors of the radial relative velocity with the existing theories. The radial relative velocity for finite Stokes number particles is governed by the interaction of particles with both large-scale energetic eddies and small-scale dissipation eddies. By assuming that both PDFs of wr and fluid velocity gradient u / x are Gaussian, Wang et al.18 developed a model for 具兩wr共R兲兩典 which combines the turbulent acceleration mechanism of the largescale turbulent motions and shear mechanism of the local small-scale dissipative motions as 具兩wr共R兲兩典 =

冑

2 2 具w 共R兲典 = r

冑

2 2 2 冑具wr,shear 典 + 具wr,accel 典. 共20兲

The acceleration mechanism36 was expressed as

冋

册 册

2 典 具wr,accel 2␥ 共1 + 2兲1/2 1 − = C w ␥−1 u ⬘2 1+

冋

0

0.1

0.2

0.3

0.4

0.5

StK

-1

10

FIG. 10. 共Color online兲 Effects of filtering operation and spectral eddy viscosity model on 具兩wr共R兲兩典 for particles at different Stokes numbers in LES 共643兲, FDNS 共kcf = 21兲, and DNS 共2563兲 flow fields. = 1.8 p / TE, Cw = 1.5 are used to fit the numerical results in Eq. 共21兲. ST in the legends denotes the ST theoretical prediction at zero Stokes number. 共a兲 具兩wr共R兲兩典 variation with Stk; 共b兲 relative error in 具兩wr共R兲兩典 of the LES and FDNS results using the DNS results as the benchmark.

FDNS

rErr of

A

rErr of

10

0

Eq.(20) ST, DNS

LES

1 1 ⫻ , 2 − 共1 + 兲 共1 + ␥兲2 / vK2 ,

1

10

2

2 具wr,shear 典

vK2

=

冉冊

1 R 15

2

共22兲

,

where R is the collision radius and was set to 0.5 in the simulation. Thus, the shear contribution scales with the small-scale flow velocity vK. The predictions of 具兩wr兩典 using Eq. 共20兲 are compared with the DNS results in Fig. 10共a兲. It is shown that Eq. 共20兲 can provide a reasonable prediction of the simulated 具兩wr兩典 data. It is noted that Eq. 共20兲 reduces to the Saffman and Turner’s 共ST兲 theory for particle with zero inertia or → 0, as 具兩wr共R兲兩典 =

冑 冑 2 R 15

,

共23兲

then 具兩wr共R兲兩典 is directly proportional to the square root of the dissipation rate in DNS, FDNS, and LES flow fields when R and are prescribed. The ST predictions of 具兩wr共R兲兩典 are 0.387, 0.285, and 0.254 for DNS, FDNS, and LES flow fields, respectively, and these are shown in Fig. 10共a兲 as well. The molecular viscosity coefficient used for the LES case is the same as the DNS case, = 0.0488. Indeed, for particles with very small Stokes numbers, the data agree with the ST predictions. This is true even for FDNS and LES when the actual dissipation rates 共see Table II兲 and molecular viscosity are used in the ST theory. The inset in Fig. 10共a兲 is an enlargement for the small StK region using linear scales, where additional data corresponding to fluid particles are added. The average radial relative velocity for fluid particles was calculated from the Eulerian velocity field using 具兩wr共R兲兩典 = 31 关具兩u共x,y,z,t兲 − u共x + R,y,z,t兲兩典 + 具兩v共x,y,z,t兲 − v共x,y + R,z,t兲兩典

共21兲

TE is the Eulerian intewhere = 2.5 p / TE, ␥ = 0.183u⬘ gral time scale, vK is the Kolmogorov velocity, and the factor Cw is used to fit the numerical results. Equation 共21兲 shows that the effect of acceleration mechanism on radial relative velocity depends on particle relative inertia , turbulent intensity u⬘2, and the turbulent Reynolds number Re since u⬘2 / vK2 = Re / 冑15. For particles with very small Stokes numbers, the relative velocity is related to the local shear rate of the flow35 2

10

+ 具兩w共x,y,z,t兲 − w共x,y,z + R,t兲兩典兴,

共24兲

where u, v, and w are the velocity along the x, y, and z axes, respectively. The numerical results of 具兩wr共R兲兩典 between fluid particles are 0.37, 0.281, and 0.238 for DNS, FDNS, and LES flow fields, respectively. Due to the loss of large amount of enstrophy in FDNS and LES flow fields, FDNS and LES both underpredict the relative velocities when compared to DNS for StK = 0. Furthermore, the relative reductions in FDNS and LES in this limit are different due to different levels of loss of enstrophy. The very minor discrepancies between our numerical results and the ST theory might come

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055106-11

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision 0.2 DNS FDNS LES

0

.

.

250 200

102

A

A

A

B C

B C

B C

150 AA

A

A

100 CBCB

B C

B C

.

10

-2

10

-1

10 StK

B

B

B

C

C

C

0

.

-0.4

B B B C C C

B C

-0.5 0

0

冑 冑 2

-0.2

-0.4

0

0.05

0.1

0.15

-0.6

10

1

10

2

0

10

(b)

1 2 v , 3 p

0.02

0.04

0.06

StK

-2

10

-1

10 StK

0

from the finite particle diameter of 0.5 assumed in the simulations since the ST theory assumes R Ⰶ . At the high Stokes number limit, one can observe in Fig. 10 that 具兩wr共R兲兩典 from DNS, FDNS, and LES are in good agreement and the relative errors of FDNS and LES approach zero. Since the velocities of the two colliding particles in this limit are completely uncorrelated and can be characterized by a Maxwellian distribution, the relative velocity is also Maxwellian. According to the analysis based on the kinetic theory for the very high Stokes number limit 共Stk → ⬁兲, 具兩wr共R兲兩典 can be expressed as33,37 具兩wr共R兲兩典 =

-0.1

-0.3

50

StK

(a)

FIG. 11. 共Color online兲 Effects of filtering operation ˙ 典 for particles and spectral eddy viscosity model on 具N c with different Stokes numbers in LES 共643兲, FDNS 共kcf = 21兲, and DNS 共2563兲 flow fields. ST in the legends denotes the Saffman and Turner’s theory for small ˙ 典 variation with St ; 共b兲 Stokes number particles. 共a兲 具N c k ˙ relative error of 具Nc典 from the LES and FDNS to the DNS results.

-0.2 rErr of

ST, DNS ST, FDNS ST, LES

rErr of

A B C

103

FDNS LES ST, FDNS ST, LES

B C

共25兲

where v2p is the variance of particle velocity. Since 具兩wr共R兲兩典 is now related to the one-point statistics v2p, which is almost unaffected by the small-scale fluid motion for particles with high Stokes numbers.14,38 Thus, the agreement of FDNS and LES with DNS at the high Stokes number limit shown in Fig. 10 are qualitatively consistent with this theoretical argument. A quantitative comparison in this limit will be provided in Sec. III E 3 when results on the collision rate are discussed. 3. Particle collision rate

Figure 11 shows the dynamically detected collision rate of particles from the DNS, FDNS, and LES. Again, the ST predictions are plotted for the small inertia limit using the dissipation rates in DNS, FDNS, and LES, respectively. The general trends from the three flow fields shown in Fig. 11共a兲 are similar. The collision rate increases very rapidly for small Stokes numbers, reaches a peak at StK = 10, and then drops slowly with an increase in StK. This behavior is consistent with the results for the collision kernel found by Sundaram and Collins33 and Zhou et al.39 The peak reflects both the effects of particle interaction with small and large scales of fluid motions. The interaction of heavy particle with smallscale motions produces a nonuniform particle distribution which enhances the collision rate through a large value of the radial distribution function at contact 具g共R兲典. On the other hand, as Stokes number increases, the turbulent transport effect which produces a large value of relative velocity, 具兩wr共R兲兩典, will dominate the collision rates.18 For small Stokes numbers, filtering causes a lower 具g共R兲典 and 具兩wr共R兲兩典, as shown in Figs. 9 and 10, and thus a lower par-

10

1

10

2

ticle collision rate according to Eq. 共18兲. The maximum relative errors occur at StK = 0.5 and are larger than 60%. For intermediate Stokes numbers, 1 ⬍ StK ⬍ 10, however, filtering causes an overestimation of 具g共R兲典 but an underestimation of 具兩wr共R兲兩典. This is a fortunate combination and leads to a rather accurate prediction of the overall collision rate. In fact, for StK ⬎ 3, both FDNS and LES yield very reasonable results for collision rate, and the difference between FDNS and LES is not noticeable. The nonmonotonic variation of the ˙ 典 for intermediate Stokes numbers relative errors in 具N c shown in Fig. 11共b兲 is a combination of the effects of 具g共R兲典 and 具兩wr共R兲兩典. In the two limiting cases of very small and very high Stokes numbers, particles uniformly distribute in the flow field, 具g共R兲典 ⬇ 1.0, the collision rate 具N˙c典 depends only on 具兩wr共R兲兩典. Based on Eqs. 共18兲 and 共23兲, the average collision rate for fluid particles 共the zero inertia limit兲 is 具N˙c典 =

冑

8 3 n20 R 15 2

冑

,

共26兲

where the average number density used in the simulations was n0 = 400 000/ 共2兲3 ⬇ 1613. Figure 11 shows that both FDNS and LES underpredict the collision rates due to the losses of dissipation rate in FDNS and LES flow fields, as implied by Eq. 共26兲. Based on Eqs. 共18兲 and 共25兲, the average collision rate for particles with very high Stokes numbers 共Stk → ⬁兲 is 具N˙c典 = 2冑n20R2冑 31 v2p .

共27兲

The quantitative comparison between the numerical collision rates and the theoretical prediction using Eq. 共27兲 for particles with high Stokes numbers is shown in Fig. 12, where ˜ = 具N˙ 典 / 共冑v2 / 3R2n2 / 2兲, the collision rate is normalized by N c

c

p

0

v2p is obtained from the simulated particle velocities in DNS, FDNS, and LES flow fields, respectively. The theoretical ˜ for high Stokes number limit is N ˜ = 4冑 = 7.09 value of N c c based on Eq. 共27兲. From Fig. 12, we observe that there is a good agreement in the collision rates obtained from DNS, FDNS, and LES. However, the theoretical value at the high Stokes number limit 共StK → ⬁兲 is still significantly above the values seen in the simulations. We note that StK = 80 for the DNS flow case is equivalent to p / TE ⬇ 5.7, where TE is the Eulerian integral time. Therefore, the theoretical limit would require a much larger p than those shown in Fig. 12. The

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8

Ñc

6 4 DNS FDNS LES Abrahamson(1975)

2 0

40

60

80

StK FIG. 12. The comparison between the normalized numerical collision rates with the kinetic theory of Abrahamson 共Ref. 37兲 for particles with high Stokes numbers.

and LES with the spectral eddy viscosity SGS model tend to underpredict the radial distribution function at contact and radial relative velocity at contact. As a result, the error on the turbulent collision rate can become quite larger. For example, at StK = 0.5, the relative error on the collision rate can be as large as 60% in LES. By examining the spectral content of the particle concentration field in DNS, we have demonstrated that the dominant scales governing the preferential concentration change with the Stokes number. Those observations further clarify the dynamics of heavy particles relevant to the SGS motions in LES. Our analysis suggests that, for StK ⬍ 3, a particle SGS model must include the effects of the SGS motions on the turbulent collision of heavy particles.

ACKNOWLEDGMENTS

difficulty to reproduce the kinetic high Stokes number limit in DNS was noted in previous studies 共for example, see Fig. 4 in Ref. 33 and Fig. 6 in Ref. 40兲. IV. CONCLUSIONS

The turbulent collision of heavy particles is a small-scale process, where the small-scale velocity field plays an important role. This represents a challenge to LES of such a process. In LES, the large-scale velocity is explicitly obtained by solving the filtered Navier–Stokes equations which are closed with an eddy viscosity SGS model such as Eq. 共6兲 to account for the effects of the SGS velocity on the large-scale one, while the SGS velocity itself cannot be obtained explicitly. The particle-pair and collision-related statistics of heavy particles are significantly affected by the SGS velocity fields. We have demonstrated here that the errors in LES of turbulent collision can be attributed to two sources: the absence of SGS velocity due to filtering and the error from the eddy viscosity SGS model. We have attempted to clearly quantify the errors from these two sources using both a priori and a posteriori tests. We carefully examine the relative errors on the radial distribution functions at contact and the radial relative velocities at contact caused by filtering and the SGS model. The relative errors of collision-related statistics increase when the filter width increases, and for particles with small Stokes numbers, the relative errors are much larger than the relative amount of turbulent kinetic energy filtered out. For a given filter width, kcf = 21 or kcf = 0.284, these relative errors are found to be quite significant for a wide range of particle Stokes numbers, as a result of the interactions of particles with small-scale fluid motions. The errors exhibit nonmonotonic dependencies on the particle Stokes numbers. For a range of intermediate Stokes numbers, such as 1 ⬍ StK ⬍ 10, the radial distribution functions at contact in the FDNS and LES can be larger than that in the DNS. This nonmonotonic variations have a fortunate consequence: the collision rate can be well predicted by the FDNS and LES when StK ⬎ 3 since the overestimation of 具g共R兲典 partly cancels the underestimation of 具兩wr共R兲兩典. For StK ⬍ 1, we find that both FDNS

This work was supported by CAS 共Grant No. KJCX2SW-L08兲, 973 Program of China 共Grant No. 2007CB8 14800兲, NSFC 共Grant Nos. 10628206, 10732090, and 10702074兲, the LNM initial funding for young investigators, and SRF for ROCS, SEM. L.-P.W. acknowledges support by the U.S. National Science Foundation 共under Grant Nos. ATM-0527140 and ATM-0730766兲 during the course of this study and by NSFC 共Grant No. 10628206兲 which made it possible for him to visit and accomplish this work at LNM, Institute of Mechanics, Chinese Academy of Sciences in China. The constructive comments from two anonymous reviewers are greatly appreciated. 1

C. T. Crowe, M. Sommerfeld, and Y. Tsuji, Multiphase Flows with Droplets and Particles 共CRC, Boca Raton, FL, 1998兲. 2 C. T. Crowe, Multiphase Flow Handbook 共CRC, Boca Raton, FL, 2006兲. 3 P. A. Vaillancourt and M. K. Yau, “Review of particle-turbulence interactions and consequences for cloud physics,” Bull. Am. Meteorol. Soc. 81, 285 共2000兲. 4 M. G. Pai and S. Subramaniam, “Modeling interphase turbulent kinetic energy transfer in Lagrangian-Eulerian spray computations,” Atomization Sprays 16, 807 共2006兲. 5 L.-P. Wang, B. Rosa, H. Gao, G.-W. He, and G. D. Jin, “Turbulent collision of inertial particles: Point-particle based, hybrid simulations and beyond,” Int. J. Multiphase Flow 35, 854 共2009兲. 6 S. C. Kassinos, C. A. Langer, G. Iaccarino, and P. Moin, Complex Effects in Large Eddy Simulations 共Springer, Berlin, 2007兲. 7 F. Yeh and U. Lei, “On the motion of small particles in a homogeneous isotropic turbulent flow,” Phys. Fluids A 3, 2571 共1991兲. 8 Q. Wang and K. D. Squires, “Large eddy simulation of particle-laden turbulent channel flow,” Phys. Fluids 8, 1207 共1996兲. 9 V. Armenio, U. Piomelli, and V. Fiorotto, “Effect of the subgrid scales on particle motion,” Phys. Fluids 11, 3030 共1999兲. 10 Y. Yamamoto, M. Potthoff, T. Tanaka, T. Kajishima, and Y. Tsuji, “Largeeddy simulation of turbulent gas-particle flow in a vertical channel: Effect of considering inter-particle collisions,” J. Fluid Mech. 442, 303 共2001兲. 11 J. G. M. Kuerten, “Subgrid modeling in particle-laden channel flow,” Phys. Fluids 18, 025108 共2006兲. 12 B. Shotorban and F. Mashayek, “Modeling subgrid-scale effect on particles by approximate deconvolution,” Phys. Fluids 17, 081701 共2005兲. 13 B. Shotorban and F. Mashayek, “A stochastic model for particle motion in large-eddy simulation,” J. Turbul. 7, 18 共2006兲. 14 P. Fede and O. Simonin, “Numerical study of the subgrid fluid turbulence

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effects on the statistics of heavy colliding particles,” Phys. Fluids 18, 045103 共2006兲. 15 A. S. Berrouk, D. Laurence, J. J. Riley, and D. E. Stock, “Stochastic modelling of inertial particle dispersion by subgrid motion for LES of high Reynolds number pipe flow,” J. Turbul. 8, 50 共2007兲. 16 M. Bini and W. P. Jones, “Large-eddy simulation of particle-laden turbulent flows,” J. Fluid Mech. 614, 207 共2008兲. 17 J. Pozorski and S. V. Apte, “Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion,” Int. J. Multiphase Flow 35, 118 共2009兲. 18 L.-P. Wang, A. S. Wexler, and Y. Zhou, “Statistical mechanical description and modelling of turbulent collision of inertia particles,” J. Fluid Mech. 415, 117 共2000兲. 19 L. I. Zaichik and V. M. Alipchenkov, “Pair dispersion and preferential concentration of particles in isotropic turbulence,” Phys. Fluids 15, 1776 共2003兲. 20 A. M. Wood, W. Hwang, and J. K. Eaton, “Preferential concentration of particles in homogeneous and isotropic turbulence,” Int. J. Multiphase Flow 31, 1220 共2005兲. 21 J.-P. Chollet and M. Lesieur, “Parameterization of small scales of threedimensional isotropic turbulence utilizing spectral closure,” J. Atmos. Sci. 38, 2747 共1981兲. 22 J.-P. Chollet, in Turbulent Shear Flows 4, edited by L. J. S. Bradbury, F. Durst, and B. E. Launder 共Springer, Berlin, 1983兲, pp. 62–72. 23 G.-W. He, R. Rubinstein, and L.-P. Wang, “Effects of subgrid-scale modeling on time correlations in large eddy simulation,” Phys. Fluids 14, 2186 共2002兲. 24 P. Fede, O. Simonin, P. Villedieu, and K. D. Squires, “Stochastic modeling of turbulent subgrid fluid velocity along inertia particle trajectories,” Proceedings of the 2006 CTR Summer Program 共Center for Turbulence Research, Stanford, CA, 2006兲. 25 E. Eswaran and S. B. Pope, “An examination of forcing in direct numerical simulations of turbulence,” Comput. Fluids 16, 257 共1988兲. 26 L.-P. Wang and M. R. Maxey, “Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence,” J. Fluid Mech. 256, 27 共1993兲.

27

Y. Yang, G.-W. He, and L.-P. Wang, “Effects of subgrid-scale modeling on Lagrangian statistics in large-eddy simulation,” J. Turbul. 9, 8 共2008兲. 28 M. R. Maxey and J. J. Riley, “Equation of motion for a small rigid sphere in a nonuniform flow,” Phys. Fluids 26, 883 共1983兲. 29 W. C. Reade and L. R. Collins, “Effect of preferential concentration on turbulent collision rates,” Phys. Fluids 12, 2530 共2000兲. 30 O. Ayala, B. Rosa, L.-P. Wang, and W. W. Grabowski, “Effects of turbulence on the geometric collision rate of sedimenting droplets: Part I. Results from direct numerical simulation,” New J. Phys. 10, 075015 共2008兲 共focus issue on cloud physics兲. 31 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids 共Oxford University Press, New York, 1987兲. 32 M. R. Maxey, “The gravitational settling of settling of aerosol particles in homogeneous turbulence and random flow fields,” J. Fluid Mech. 174, 441 共1987兲. 33 S. Sundaram and L. Collins, “Collision statistics in an isotropic particleladen turbulent suspension. Part I: Direct numerical simulations,” J. Fluid Mech. 335, 75 共1997兲. 34 H. Yoshimoto and S. Goto, “Self-similar clustering of inertial particles in homogeneous turbulence,” J. Fluid Mech. 577, 275 共2007兲. 35 P. G. Saffman and J. S. Turner, “On the collision of drops in turbulent clouds,” J. Fluid Mech. 1, 16 共1956兲; “Corrigendum,” ibid. 196, 599共E兲 共1988兲. 36 F. E. Kruis and K. A. Kusters, “The collision rate of particles in turbulent flow,” Chem. Eng. Commun. 158, 201 共1997兲. 37 J. Abrahamson, “Collision rates of small particles in a vigorously turbulent fluid,” Chem. Eng. Sci. 30, 1371 共1975兲. 38 G.-D. Jin, G.-W. He, L.-P. Wang, and J. Zhang, “Subgrid scale fluid velocity timescales seen by inertial particles in large-eddy simulation of particle-laden turbulence,” Int. J. Multiphase Flow 36, 432 共2010兲. 39 Y. Zhou, A. S. Wexler, and L.-P. Wang, “On the collision rate of small particles in isotropic turbulence: II. Finite inertia case,” Phys. Fluids 10, 1206 共1998兲. 40 Y. Zhou, A. S. Wexler, and L.-P. Wang, “Modelling turbulent collision of bidisperse inertial particles,” J. Fluid Mech. 433, 77 共2001兲.

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Large-eddy simulation of turbulent collision of heavy particles in isotropic turbulence Guodong Jin,1 Guo-Wei He,1,a兲 and Lian-Ping Wang1,2 1

State Key Laboratory for Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 2 Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716-3140, USA

共Received 1 October 2009; accepted 30 March 2010; published online 20 May 2010兲 The small-scale motions relevant to the collision of heavy particles represent a general challenge to the conventional large-eddy simulation 共LES兲 of turbulent particle-laden flows. As a first step toward addressing this challenge, we examine the capability of the LES method with an eddy viscosity subgrid scale 共SGS兲 model to predict the collision-related statistics such as the particle radial distribution function at contact, the radial relative velocity at contact, and the collision rate for a wide range of particle Stokes numbers. Data from direct numerical simulation 共DNS兲 are used as a benchmark to evaluate the LES using both a priori and a posteriori tests. It is shown that, without the SGS motions, LES cannot accurately predict the particle-pair statistics for heavy particles with small and intermediate Stokes numbers, and a large relative error in collision rate 共up to 60%兲 may arise when the particle Stokes number is near StK = 0.5. The errors from the filtering operation and the SGS model are evaluated separately using the filtered-DNS 共FDNS兲 and LES flow fields. The errors increase with the filter width and have nonmonotonic variations with the particle Stokes numbers. It is concluded that the error due to filtering dominates the overall error in LES for most particle Stokes numbers. It is found that the overall collision rate can be reasonably predicted by both FDNS and LES for StK ⬎ 3. Our analysis suggests that, for StK ⬍ 3, a particle SGS model must include the effects of SGS motions on the turbulent collision of heavy particles. The spectral analysis of the concentration fields of the particles with different Stokes numbers further demonstrates the important effects of the small-scale motions on the preferential concentration of the particles with small Stokes numbers. © 2010 American Institute of Physics. 关doi:10.1063/1.3425627兴 I. INTRODUCTION

Particle-laden turbulent flows are encountered in a wide range of engineering and environmental problems;1,2 for example, pollutant or aerosol particle dispersion, warm rain droplet formation in the atmosphere, and fluidization and turbulent mixing in combustion processes.3,4 Understanding the hydrodynamics of such flows is the basis for many applications in engineering design, environmental protection, and weather forecasting. Particularly, turbulence-mediated collision of heavy particles is vital to efficiency and conversion rates in these processes.5 Large-eddy simulation 共LES兲 has emerged as a promising tool for simulating turbulent flows in general and, in recent years, has also been applied to particle-laden turbulence with some successes.6 The motion of heavy particles is much more complicated than fluid elements, and therefore, LES of turbulent flow laden with heavy particles encounters new challenges. In LES, only large-scale eddies are explicitly resolved and the effects of unresolved, small-scale or subgrid scale 共SGS兲 eddies on the large-scale eddies are parametrized. The SGS turbulent velocity field is not available. a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected] and [email protected] Telephone: 86-1082543969. Fax: 86-10-82543408.

1070-6631/2010/22共5兲/055106/13/$30.00

The effects of SGS turbulent velocity fields on particle motions have been studied in channel flows or isotropic turbulent flows by Yeh and Lei,7 Wang and Squires,8 Armenio, Piomelli, and Fiorotto,9 Yamamoto et al.,10 Kuerten,11 Shotorban and Mashayek,12,13 Fede and Simonin,14 Berrouk et al.,15 Bini and Jones,16 Pozorski and Apte,17 among others. The Eulerian deconvolution method11,12 or Lagrangian stochastic approach13,15,17 is usually used to model the effects of SGS motions on particle dispersion. The missing of SGS turbulent velocity does not pose a serious problem for quantifying the properties that are mainly governed by large-scale turbulent eddies, i.e., single-particle statistics such as oneparticle dispersion coefficient.7–9 However, the turbulent collision of heavy particles is a small-scale process, where the small-scale velocity field plays an important role. The collision-related relative motions and pair distribution of heavy particles with the Stokes numbers 共StK ⬅ p / K, i.e., the ratio of heavy particle response time to flow Kolmogorov time兲 on the order of 1 is mainly determined by the smallscale eddies.18–20 These particle-pair statistics and thus collision rate of heavy particles are unlikely to be properly simulated in LES. This is one of the main challenges in LES of particle-laden turbulent flows. Another issue is that the flow field in LES is more coherent in space and more correlated in time than either the fully resolved flow field or direct nu-

22, 055106-1

© 2010 American Institute of Physics

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Phys. Fluids 22, 055106 共2010兲

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merical simulation 共DNS兲 one due to the filtering operation and overdissipation of the commonly used eddy viscosity SGS model.21–23 The flow field from the LES with an eddy viscosity SGS model may not recover the properties especially needed for collision-rated statistics of heavy particles. Fede and Simonin14 studied the effects of SGS turbulent motions on particle-pair statistics of nonsettling, monodispersed heavy particles in an isotropic turbulent flow field. DNS was used to generate a turbulent flow at Taylor microscale Reynolds number Re = 34.1. The filtered-DNS 共simply denoted as FDNS hereafter兲 flow field was obtained from the DNS one by filtering the small-scale fluid motions at different cutoff scales. One-way coupling was assumed and particles were treated as perfect-elastic hard spheres. They focused on the effects of SGS motions on both one- and twoparticle statistics of heavy particles. They concluded that the effects of SGS motions must be included when the SGS Stokes number p / ␦ [email protected] is less than 5.0, where ␦ [email protected] is the SGS Lagrangian integral time scale seen by heavy particles. A very interesting result is that the effect of filtering leads to a reduction in preferential concentration if p ⬍ 0.5␦ [email protected], but an increase in the preferential concentration if p ⬎ 0.5␦ [email protected] For the case of p ⬍ 0.5␦ [email protected], the eddies at the dissipation scales play an important role in particle clustering. Filtering reduces the intensity of the dissipation-scale eddies and thus reduces the level of preferential concentration. For the case of 5␦ [email protected] ⬎ p ⬎ 0.5␦ [email protected], the particles respond to the eddies ranging from the dissipation-scale eddies to the larger-scale ones. By removing the dissipationscale eddies, the larger-scale eddies may actually become more coherent in space and better correlated in time, although the maximum vorticity intensity is reduced. This implies that the actual level of preferential concentration is governed not only by the vorticity intensity but also by the vorticity distribution. Fede et al.24 then developed a onepoint stochastic Lagrangian model to recover the SGS motions seen by heavy particles. The model predicts a good match to DNS data for the particle kinetic energy. The collision-related statistics has not been quantitatively compared and assessed simultaneously using DNS, FDNS, and LES in previous studies. The aim of this paper is to investigate the effects of the spectral eddy viscosity SGS model and the SGS fluid motions on particle-pair statistics such as radial distribution function at contact, radial relative velocity at contact, and collision rate. By understanding these effects, we hope to develop some insights into the dynamic interaction of heavy particles with SGS flow fields, as well as a guidance on a better particle SGS model aimed at improving particle-pair statistics in LES. This paper is organized as follows. An overview of the governing equations and simulation methods is given in Sec. II The numerical results about the effects of SGS motions on collision-related statistics of heavy particles are presented in Sec. III. Conclusions on LES of turbulent collision of heavy particles can be found in Sec. IV.

II. GOVERNING EQUATIONS AND SIMULATION METHODS

The equations governing fluid flow and motion of heavy particles are described in this section, along with our numerical methods. A. DNS method

The Navier–Stokes equations for isotropic turbulence are

冉

冊

u p 1 = u ⫻ − ⵜ + u2 + ⵜ2u + f共x,t兲, t 2

共1兲

ⵜ · u = 0,

共2兲

where u denotes the velocity, = ⵜ ⫻ u is the vorticity, p is the pressure, is the fluid density, and is the fluid kinematical viscosity. The flow is driven and maintained by a random forcing f共x , t兲, which is nonzero only at low wavenumbers in Fourier space 兩k兩 ⬍ 冑8.25,26 The DNS of isotropic turbulence is performed using a standard pseudospectral method on N3 grids covering a periodic box of side L = 2. Here, N = 256. In Fourier space, Eqs. 共1兲 and 共2兲 can be represented as 共k ⱕ kmax兲,

冉

冊

+ k2 uˆ 共k,t兲 = P共k兲F共u ⫻ 兲 + ˆf共k,t兲, t

共3兲

where uˆ 共k , t兲 is the Fourier coefficient or the fluid velocity in Fourier space, F denotes a Fourier transformation, the projection tensor P = 兵P jm其 = ␦ jm − k jkm / k2 共j , m = 1 , 2 , 3兲 projects F共u ⫻ 兲 onto the plane normal to k and eliminates the pressure term in Eq. 共1兲 with the aid of the continuity equation 共2兲, and kmax is the maximum wavenumber. The wavenumber components in Fourier space are k j = n j共2 / L兲, where n j = −N / 2 , . . . , −1 , 0 , 1 , . . . , N / 2 − 1 for j = 1 , 2 , 3. Aliasing errors are removed using the two-thirds truncation method. The spatial resolution is monitored by the value of kmax, where is the Kolmogorov length scale. The value of kmax is typically larger than 1.1 in the present simulation. The Fourier coefficients are advanced in time using a second-order Adams–Bashforth method for the nonlinear term and an exact integration for the linear viscous term. The time step is chosen to ensure that the Courant– Friedrichs–Lewy number is less than 0.5 for numerical stability and accuracy.26,27 B. FDNS velocity fields

The FDNS velocity field is obtained from the DNS velocity field by truncating the Fourier coefficients larger than the cutoff wavenumber kcf, kcf

˜u共x,t兲 =

兺

uˆ 共k,t兲eik·x ,

共4兲

兩k兩=k0

where ˜u共x , t兲 is the filtered velocity in physical space and kcf is the cutoff wavenumber in FDNS. k0 = 1 is the lowest wavenumber. FDNS can be regarded as an ideal LES to study the

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effects of SGS eddies on the statistics of particle motions since FDNS does not contain any eddy viscosity SGS model errors.14 C. LES method

The LES of isotropic turbulence is performed on the coarser grids using the same pseudospectral method and large-scale forcing scheme as DNS. The governing equation in LES is given by 共兩k兩 ⱕ kc, where kc is the maximum wavenumber in LES兲

再

冎

+ 关 + e共k兩kc兲兴k2 uˆ共k,t兲 = P共k兲F共u ⫻ 兲 + ˆf共k,t兲, t

TABLE I. The parameters in DNS and LES. DNS 共2563兲

LES 共643兲

Reynolds number rms fluid velocity u⬘ 共Effective兲 dissipation rate Minimum length scale

102.05 19.34

¯ 18.52

3771.4 0.0135

3434.8 0.0238

Minimum time scale

0.0037

0.0055

Minimum velocity scale

3.6272

4.327

Eulerian integral time scale Molecular viscosity

0.050 0.0488

0.056 0.0488

Method

共5兲 where u and are the resolved velocity and vorticity in physical space, respectively. The term e共k 兩 kc兲k2uˆ共k , t兲 on the left-hand side models the net effects of SGS motions on the resolved motions. The spectral eddy viscosity SGS model21,22 is used in this paper, given by

冑

E共kc兲 , kc

共6兲

with

+e 共k/kc兲 = CK−3/2关0.441 + 15.2 exp共− 3.03kc/k兲兴.

共7兲

The spectral viscosity e共k 兩 kc兲 depends on the wavenumber k, the maximum wavenumber kc as well as E共kc兲, the value of the energy spectrum function at kc. Here, E共kc兲 in Eq. 共6兲 is dynamically evaluated from the LES fluid field. CK = 2.5 is taken in this paper. D. Motion of inertial particles

The dispersed phase is composed of N p spherical heavy particles with a diameter smaller than the Kolmogorov length scale , d p = 0.5. In general, there are several different forces acting on a heavy particle suspended in a nonuniform and unsteady turbulent flow field.28 When the ratio of particle density to fluid density is much larger than 1 共 p / f Ⰷ 1兲, the equation of motion for a heavy particle can be approximated as26 dx p共t兲 = v p共t兲, dt

共8兲

dv p共t兲 u关x p共t兲,t兴 − v p共t兲 + w0 , = dt p

共9兲

III. NUMERICAL RESULTS A. Statistics of DNS and LES flow fields

Table I lists Eulerian statistics of the DNS and LES flow fields in this study. The root mean square 共rms兲 of turbulent fluctuation velocity u⬘ and the average dissipation rate are computed from the three dimensional turbulent energy spectrum function shown in Fig. 1, u⬘ =

冑

1 具uiui典 = 3

冑冕 2 3

kmax

共10兲

E共k兲dk,

k0

2

10

where x p共t兲 and v p共t兲 are the particle position and velocity at time t, w0 is the particle terminal or settling velocity under gravity in otherwise quiescent fluid, w0 = g p, p is the particle Stokes response time, and g is the gravitational acceleration. The particle terminal velocity is set to w0 = 共−vK , 0 , 0兲 in this study, where vK is the Kolmogorov velocity obtained from DNS. The linear Stokes drag force is assumed due to the low particle Reynolds number, u关x p共t兲 , t兴 is the fluid velocity seen by a heavy particle which is obtained from the DNS, FDNS, and LES flow fields, respec-

101 E(k)

e共k兩kc兲 = +e 共k/kc兲

tively, by a six-point Lagrangian interpolation scheme in each direction.27 The equation of motion 关Eq. 共9兲兴 is integrated with a fourth-order Adams–Bashforth method for the particle velocity and then a fourth-order Adams–Moulton method for the particle location in Eq. 共8兲. The particles are assumed to have identical size and density, and each moves independent of others 共i.e., the so-called ghost particle model兲. This is appropriate for turbulent geometric collision.29,30 All particle pairs and collision search are conducted using the efficient cell-index method and the concept of linked lists.18,31

100

DNS 2563 LES 643 k-5/3

-1

10

10-2 0 10

1

10 k

2

10

FIG. 1. The energy spectra of the simulated flows in 2563 DNS and 643 LES.

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Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang

冕

kmax

2k2E共k兲dk.

共11兲

k0

In LES, the molecular viscosity coefficient in Eq. 共11兲 is replaced by the effective viscosity + ve共k 兩 kc兲 to calculate the effective dissipation rate, e =

冕

kc

2关 + ve共k兩kc兲兴k2E共k兲dk,

共12兲

k0

which represents the sum of the molecular dissipation rate of large-scale modes and the dissipation rate of the SGS eddies. The dissipation rate in LES 共643兲 listed in Table I is the effective one. The actual or molecular dissipation rate of the large-scale motion is 1626.6. As the spectral eddy viscosity coefficient ve共k 兩 kc兲 varies with the wavenumber, an average SGS viscosity coefficient can be estimated using the enstrophy spectrum 2k2E共k兲 as weight factors, ¯SGS =

兰kkc 2ve共k/kc兲k2E共k兲dk 0

兰kkc 2k2E共k兲dk

共13兲

.

0

This average eddy viscosity coefficient in LES, ¯SGS = 0.0543, is only slightly larger than the molecular viscosity coefficient used in DNS, = 0.0488. This could be due to the moderate flow Reynolds number realized in DNS, given that the same large-scale flow forcing and flow domain size are applied in LES and DNS. The smallest 共Kolmogorov兲 length, time, and velocity scales in DNS are

= 共3/兲0.25,

K = 共/兲0.5,

vK = 共兲0.25 .

共14兲

In order to obtain the minimum characteristic scales in LES flow field, the effective viscosity 共 + ¯SGS兲 and effective dissipation rate e are used to replace the molecular viscosity and molecular dissipation rate , respectively, in Eq. 共14兲 to represent the smallest characteristic length, time, and velocity scales as

e =

冋

共 + ¯SGS兲3 e

册

0.25

,

K,e =

冋

共 + ¯SGS兲 e

册

0.5

,

vK,e = 关e共 + ¯SGS兲兴0.25 .

共15兲

Ideally, e = if there was no SGS model error in LES. Figure 1 plots the resulting energy spectra of DNS and LES flow fields in log-log coordinates. It is shown that the energy spectrum of LES flow field decays faster than that of DNS flow field at high wavenumbers. This is due to the overdissipation in the spectral eddy viscosity SGS model. In this study, the cutoff wavenumber in FDNS is taken as kcf = 21, corresponding to the LES at the grid resolution of 643. B. Preferential concentration of heavy particles

It is well known that, unlike passive tracers, heavy particles are distributed nonuniformly in a turbulent flow field due to their interaction with vortical and straining motions of local flows. Maxey32 developed an asymptotic analysis of the preferential concentration of particles with weak inertia. If

we consider the particle distribution from an Eulerian viewpoint and the particle velocity field is then defined as v共x , t兲, its divergence field is ⵜ · v = − p

冉

冊

u j ui 1 = − p s2i,j − 2 , 2 xi x j

共16兲

where = 兩ⵜ ⫻ u兩 and sij = 21 共ui / x j + u j / xi兲 are the local fluid vorticity and strain rate tensor, respectively. Equation 共16兲 indicates that particle velocity field is compressible even in an incompressible fluid velocity field and particles accumulate in regions of low vorticity and high strain rate. Figure 2 shows the preferential concentration of heavy particles at different particle Stokes numbers in a DNS flow field. N p = 1.2⫻ 106 heavy particles are randomly released into a statistically stationary flow field. The spatial distribution of particles evolves with time and particle preferential concentration develops in the flow field. After about four times of the Eulerian integral time scale, the accumulation is balanced by randomly turbulent mixing and the preferential concentration reaches its quasiequilibrium state. The particle terminal velocity is w0 = 共−vK , 0 , 0兲 for all kinds of particles, where vK is the Kolmogorov velocity. The background shows the contour of local flow vorticity at z = obtained from DNS 共2563 , Re = 102.05兲. The locations of all particles in a slice with a thickness of 2 / 256 centered at z = are projected onto the x-y plane. Three Stokes numbers are considered: StK = 0.1, 1.0, 10.0. The degree of nonuniformity depends on the particle Stokes numbers. For the Stokes number StK = 0.1, the heavy particle response time is much shorter than the Kolmogorov time scale; therefore, particles respond quickly to the changes of the flow field and nearly follow the fluid motion. In this case, large-scale fluid motion serves to efficiently disperse the particles. However, even at StK = 0.1, the inertial bias is noticeable and more particles are found in the regions of low flow vorticity 关see the dark regions in Fig. 2共a兲兴. For StK = 1, particle motion is strongly affected by small-scale eddies and particles are centrifuged out of the vortical structures, leading to a strong nonuniformity 关see Fig. 2共b兲兴. As heavy particle response time increases, particles respond to the eddies with larger time scales relative to the Kolmogorov eddies. The level of accumulation starts to drop when StK ⬎ 1. At StK = 10, particles accumulate into large-scale structures and are mixed by small eddies. Therefore, particles return back to a more uniform distribution 关see Fig. 2共c兲兴. Since the particles have a fixed terminal velocity of vK in the negative x direction, patches of particles aligned in the vertical direction are visible in Figs. 2共b兲 and 2共c兲 as a result of the preferential sweeping.26 In general, the particle concentration field contains more large-scale structures as the Stokes number increases. This is demonstrated in Fig. 3, which plots the energy spectra of concentration fluctuations at the Stokes numbers StK = 0.1, 0.5, 1.0, 3.0, 5.0, and 10.0, ranging from small to large inertial particles. The particle concentration C共x , t兲 is defined as the number of particles in a local cell of center x. In the present calculations, 1283 coarse grids are used to smooth the particle concentration field, although 2563 fine grids are used

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055106-5

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision ω 6 5 4 3 2 1

6 5

5 4

g x

x

4 3

2

1

1 1

2

3 y

4

5

g

3

2

0 0

ω 6 5 4 3 2 1

6

0 0

6

1

(a) StK= 0.1

2

4

5

6

(b) StK = 1.0 ω 6 5 4 3 2 1

6 5 4 x

3 y

g

3 2 1 0

0

1

2

3 y

4

5

6

(c) StK = 10.0 FIG. 2. 共Color兲 The preferential concentration of particles with different Stokes numbers in a slice centered at z = with a thickness of 2 / 256. The background is the vorticity contour at z = of DNS flow field, which is normalized by the average vorticity. The arrow below the legend represents the direction of gravity and the particle terminal velocity is set to w0 = 共−vK , 0 , 0兲, where vK is the Kolmogorov velocity. For a small Stokes number in 共a兲, the particles almost follow the trajectories of fluid particles and uniformly distribute in the flow field. For a large Stokes number in 共c兲, the particles are too heavy to respond to the eddies and also uniformly distribute in the flow field. For an intermediate Stokes number in 共b兲, the preferential concentration is most obvious.

for the flow field in DNS. The difference between the particle concentration C共x , t兲 at time t and the initial concentration C共x , 0兲 is defined as the concentration fluctuations, which largely remove the statistical noises in practical computations. Since the probability distribution function 共PDF兲 of the initial concentration C共x , 0兲 is set to be a Poisson distribution, the difference C共x , t兲 − C共x , 0兲 measures the fluctuations of concentration C共x , t兲 relative to the uniform concentration C共x , 0兲. When the particle concentration field becomes statistically stationary, the concentration fluctuations are trans-

formed into the Fourier space and the squared magnitudes of the modes in each wavenumber shell within the two radii between k − 0.5 and k + 0.5 are summed to yield the concentration energy spectrum Ec共k兲, where k = 1 , 2 , 3 , . . .. Ten time frames with a time interval equal to 0.2TE are used to obtain an average energy spectrum for each Stokes number, where TE is the Eulerian integral time. The area under each curve is equal to 具C2典 − 具C2共0兲典 = 兰⬁1 Ec共k兲dk, where 具C2共0兲典 denotes the concentration variance of heavy particles at initial time. It measures the degree of nonuniformity of the particle con-

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055106-6

6 5

-2

6 5

4

Ec(k)

10

3 2

-4

1

10-5

(a)

5

6

3

2 4 5

2

6

1 2 3 4 5 6

100

Stk=0.1 Stk=0.5 Stk=1.0 Stk=3.0 Stk=5.0 Stk=10.0

4 3

0.01

1

6

4 3

0.005

6

3 3

102

(b)

FIG. 3. 共Color online兲 Wavenumber spectra of particle concentration fields of heavy particles with different Stokes numbers. 共a兲 Logarithmic scales; 共b兲 linear scales.

3

2 4 2 4

5

2

0

Stk=0.1 Stk=0.5 Stk=1.0 Stk=3.0 Stk=5.0 Stk=10.0

3 5

6 2

2 1

101 k

1 2 3 4 5 6

4

5

1

1

10

3 24 6

2 -3

0.015

4 5 6 3 5 3 6 2 2

Ec(k)

10

Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang

6 1

5

20

40

60

k

centration for a given Stokes number. This area increases with Stk, reaches a maximum at Stk = O共1兲, and then decreases with Stk. It is observed from Fig. 3共a兲 that as the Stokes numbers increase, the peaks of concentration spectra move from larger wavenumbers to smaller ones with more concentration energy contained at small wavenumbers. The peak wavenumber gives an indication of the length scale at which these eddies contribute most significantly to the concentration field. Figure 3共b兲 shows that the small scales contain more concentration energy than the large ones for StK ⱕ 3, while the small scales contain less concentration energy than the large ones for StK ⬎ 3. Therefore, Fig. 3 implies that the small-scale eddies are important to the particle concentration for StK ⱕ 3, and simply neglecting small-scale eddies in the conventional LES may lead to incorrect predictions. A particle SGS model to account for the effects of fluid motions at small scales on particle motions is needed in LES. In order to study the effects of average particle number concentration on the energy spectrum of particle concentration field, we doubled the number of particles to N p = 2.4 ⫻ 106 while keeping the same mesh resolution, i.e., 1283 when defining the local number concentration. The energy spectra of nondimensional particle concentration fields, Eˆc共k兲, are plotted in Fig. 4, where the nondimensional particle concentration field is defined as C共x , t兲 / 具C典 and 具C典 = N p / 共2兲3 is the volume-averaged particle concentration. In Fig. 4, the thin solid lines with symbols are the results obtained with N p = 1.2⫻ 106 and the thick dashed lines are the

results obtained with N p = 2.4⫻ 106. We observe that the wavenumber spectra of the nondimensional particle concentration are not sensitive to N p for all Stokes numbers considered, although the curves obtained with the larger N p are smoother as expected. C. Radial distribution function

The nonuniformity of particle concentration shown in Fig. 2 has an important consequence on turbulent collision rates, which can be quantified in terms of the radial distribution function.18,33 For monodispersed particle system in a statistically isotropic flow field, the radial distribution function g共r兲 is defined as the ratio of the number density of particle pairs with interparticle distance from r to r + dr, Npair共r兲 / Vs, to the average number density of particle pairs in the flow domain, g共r兲 =

Npair共r兲 Vs

冒

1 2 N p共N p

共2兲

− 1兲 3

共17兲

,

where Vs = 34 关共r + dr兲3 − r3兴 is the volume of the spherical shell, Npair共r兲 is the number of particles in the shell, and N p共N p − 1兲 / 2 is the total possible particle pairs for N p monodispersed particles.18 For a uniformly distributed system, the radial distribution function approaches 1. Figure 5 shows the transient variation of the radial distribution function of particles with StK = 1.0 in the 2563 DNS flow field. At beginning, the particles are uniformly distributed so that g共r / R兲 = 1, where R is

K=10.0 0.04 StSt K=5.0

StK=3.0

101

0.03 0.02

g(r/R)

Êc(k)

StK=1.0

10

StK=0.5

0.01

0

t=0.0 t=0.025 t=0.05 t=0.1 t=0.2 t=0.25

StK=0.1

0 0

20

40

60

k FIG. 4. 共Color online兲 Wavenumber spectra of nondimensional particle concentration. The thin solid lines with symbols are the results obtained with N p = 1.2⫻ 106 and the thick dashed lines are the results with N p = 2.4⫻ 106. The wavenumber spectra for the two particle numbers almost collapse for each Stokes number considered.

10

-1

1

3 r/R

5

7

9

FIG. 5. Variation in the radial distribution function g共r / R兲 with the separation distance at different times in flow field of 2563 DNS, where the particle Stokes number StK = 1.0. g共r / R兲 exhibits a power-law scaling with r.

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055106-7

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision

15

result from the random large-scale forcing. The time averaged value of g共R兲 does not depend on the used particle number. The results reported in the following are based on 4 ⫻ 105 particles and are averaged over time after the quasiequilibrium state is obtained.

g(R)

10

D. Effects of filter width on the collision-related statistics

5

The collision rate of monodispersed particles is formulated as18

0

0

2

4

t/TE

6

8

10

FIG. 6. Variation in the radial distribution function at contact g共R兲 with time in flow field of 2563 DNS and the particle Stokes number StK = 1.0. The flow reaches a statistically steady state with fluctuations after about four times of the Eulerian integral time scale, TE = 0.05. The fluctuation is related to the finite numbers of particles, which is shown in Fig. 7.

the geometric collision radius, R = d p for monodispersed particles, and d p is the particle diameter. As time increases, particle concentration deviates from the initially uniform distribution and g共r / R兲 increases monotonically with time. Finally, the concentration field approaches a statistically stationary state after about four times of the Eulerian integral time scale TE. At the quasiequilibrium, we found that g共r / R兲 follows a power-law scaling with r / R, consistent with previous results.29 It is also observed in Fig. 5 that g共r / R兲 still follows the power-law scaling even in the case of sedimentation. For geometric collision rate, the radial distribution function at contact, i.e., g共r / R = 1兲, is of interest.33 Figure 6 shows the variation in g共r / R = 1兲 with time. At long times, g共r / R兲 fluctuates with time due to the finite number of particles and turbulent mixing. The magnitude of fluctuation decreases as the particle number increases. Figure 7 shows that the magnitude of fluctuation for 106 particles is smaller than that of 4 ⫻ 105 particles in the 643 LES flow field where the particle Stokes number StK = 1.3. Some of the fluctuations

10

g(R)

8 6

5

4x10 particles 1x106 particles

4 2 0 0

2

4

6

8

t/TE FIG. 7. The effect of particle numbers on the magnitude of fluctuation of g共R兲 in the flow of 643 LES, StK = 1.3. The fluctuation magnitude of g共R兲 for 106 particles is much smaller than that for 4 ⫻ 105. The Eulerian integral time scale of LES flow field TE = 0.056.

2

n 具N˙c典 = 2R2具兩wr共R兲兩典具g共R兲典 0 , 2

共18兲

where n0 = N p / 共2兲3 is the average particle number density, N p is the total particle number in the flow domain, 具兩wr共R兲兩典 is the averaged radial relative velocity of collision pairs at contact, 具兩wr共R兲兩典 =

冓冏 冏冔 w共R兲 · r r

,

共19兲

where r is the vector connecting the centers of the collision pair, r = 兩r兩, and w共R兲 = v p1 − v p2 is the relative velocity between the two particles. The angle brackets denote an ensemble average over all collision pairs. In order to study the effects of different filter widths on particle collision-related statistics, we calculate 具g共R兲典, 具兩wr共R兲兩典, and 具N˙c典 of particles with different Stokes numbers StK = 0.5, 1, and 5 in different FDNS flow fields with the cutoff wavenumbers kcf = 64, kcf = 42, and kcf = 21, respectively. The statistics of the energy, enstrophy, and energy dissipation rate in 2563 DNS, FDNS, and 643 LES flow fields are given in Table II. As the energy spectrum of FDNS for k ⱕ kcf is the same as that of the DNS, typically very little turbulent kinetic energy is contained within high wavenumbers 共k ⬎ kcf兲. The percentage of energy filtered out in the FDNS is negligible except that the cutoff wavenumber is sufficiently low. For the largest filter width at kcf = 21 in the present study, only 3% of the kinetic energy is filtered out, so the rms turbulent velocity in the FDNS flow field differs very little from that of the DNS flow field. However, as enstrophy is defined as 兰kkcf2k2E共k兲dk, a large portion of enstrophy is 0 distributed at high wavenumbers, thus a large percentage of enstrophy is filtered out in FDNS. For the case of kcf = 21, about 45.7% of enstrophy is filtered out. With the same spatial resolution in the FDNS 共kcf = 21兲 and 643 LES, the total turbulent energy of LES is lower than that of FDNS, and the enstrophy is even lower due to the overdissipation of the eddy viscosity SGS model. The relative errors in 具g共R兲典, 具兩wr共R兲兩典, and 具N˙c典, using DNS results as a benchmark, are shown in Fig. 8. For a given Stokes number, the relative errors of these statistics increase with increasing filter width, and the relative errors for 具g共R兲典, 具兩wr共R兲兩典, and 具N˙c典 are not the same. For example, when kcf = 21 and StK = 0.5, the relative errors for 具g共R兲典, 具兩wr共R兲兩典, and 具N˙c典 are ⫺0.37, ⫺0.29, and ⫺0.57, respectively. For a given filter width, the filter operation has different effects on the motion of particles with

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055106-8

Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang TABLE II. Kinetic energy, enstrophy in DNS 共2563兲, FDNS, and LES 共643兲. Method

DNS 共2563兲

kcf = 64

kcf = 42

kcf = 21

kcf

1.15

0.864

0.567

0.285

¯

561.3 77 290.6

561.0 74 409.8

559.5 66 833.7

544.2 41 994.6

514.5 33 330.6

Actual dissipation rate

3771.4

3631

3261

2049

1626.6

Energy filtered out 共%兲

¯

0.05

0.32

3.04

8.3

Enstrophy filtered out 共%兲

¯

3.72

13.5

45.7

56.9

Energy Enstrophy

different Stokes numbers. The reason is that particles with different inertia are related to different length scales of eddies.34 It implies that filtering operation can result in a higher or lower prediction of 具g共R兲典 for different Stokes ˙ 典. It numbers and always lower prediction of 具兩w 共R兲兩典 and 具N r

c

0.1

0.1

0

0

-0.1

StK=0.5

-0.2

StK=1.0 StK=5.0

rErr of

rErr of

is important to note that, for small Stokes numbers, the relative error on the relative velocity is much larger than the relative amount of turbulent kinetic energy removed, since the main contribution to the particle relative motion comes from small-scale fluid motion. Our simulation results about the effects of filtering on particle collision rate are consistent with those of Fede and Simonin.14 We have included the data in Fig. 28 of Fede and Simonin14 in Fig. 8共c兲 for comparison. Some of the quantitative difference between their data and ours could be due to the fact that Fede and Simonin14 assumed zero settling and the Stokes numbers are not the same. The effects of inertia on particle-pair statistics will be discussed in detail later.

-0.3 -0.4 0.2 (a)

E. Effects of eddy viscosity SGS model on collision-related statistics

A spectral eddy viscosity SGS model 关see Eqs. 共6兲 and 共7兲兴 is used to close the filtered Navier–Stokes equations in LES. From Fig. 1, one can see that the spectral eddy viscosity SGS model overdissipates the turbulent kinetic energy, especially near the cutoff wavenumbers, making the vorticity field in LES more coherent than that in FDNS with the same spatial resolution. As particle turbulent collision rate is related to the particle preferential concentration and the averaged radial relative velocity, we will first study the influences of SGS motions and the spectral eddy viscosity SGS model on the two aforementioned quantities and then the collision rates using both a priori and a posteriori tests, i.e., we will compare the results from the DNS and the ones from the FDNS and the LES flow fields, respectively. This study will be carried out over a wide range of the Stokes numbers.

-0.1

StK=0.5

-0.2

StK=1.0 StK=5.0

-0.3 0.4

LES 共643兲

0.6 0.8 kcf/kmax

-0.4 0.2

1

0.4

(b)

0.2

rErr of

0 .

StK=1.0 StK=5.0 StK=0.37 StK=0.88 StK=4.73

-0.4

(c)

1

StK=0.5

-0.2

-0.6 0.2

0.6 0.8 kcf/kmax

FIG. 8. Effects of filter width on the collision-related statistics of different Stokes numbers, where rErr denotes the relative error of the result from FDNS to that from DNS, rErr= 共␣FDNS − ␣DNS兲 / ␣DNS, and ␣ represents a statistical quantity and the open symbols in 共c兲 are our numerical results and the solid symbols are obtained from Fig. 28 of Fede and Simonin 共Ref. 14兲. 共a兲 The radial distribution function at contact 具g共R兲典; 共b兲 the radial relative velocity at contact 具兩wr共R兲兩典; 共c兲 the ˙ 典. particle collision rate 具N c

0.4

0.6 0.8 kcfη

1

1.2

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055106-9

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision

13 0.4

DNS FDNS

9

rErr of

LES

5

FDNS LES

0.2

FIG. 9. Effects of filtering operation and spectral eddy viscosity model on 具g共R兲典 of different Stokes numbers in LES 共643兲, FDNS 共kcf = 21兲, and DNS 共2563兲 flow fields. 共a兲 具g共R兲典 variation with Stk; 共b兲 relative error of 具g共R兲典 from LES and FDNS to the DNS results.

0

-0.2 -0.4

1 10-2

(a)

10-1

100 St K

101

10-2

102

(b)

10-1

100 StK

1. Radial distribution function

A nonuniform distribution of particles 共the accumulation effect兲 at the Stokes numbers near one is mainly induced by small-scale eddies which are directly related to the cutoff wavenumber in FDNS and related to both the cutoff wavenumber and the spectral eddy viscosity SGS model in LES. For kcf = 21, the space resolution in 2563 DNS is four times of the ones in FDNS and LES 共⌬xLES = ⌬xFDNS = 4⌬xDNS兲. Figure 9共a兲 shows the dependence of 具g共R兲典 from the DNS, FDNS, and LES on the Stokes numbers, and Fig. 9共b兲 plots the relative errors 共rErr兲 of the FDNS and LES to the DNS results. Figure 9 shows that there are three regimes for 具g共R兲典. In the limit cases of both StK → 0 and StK → ⬁, particles tend to be uniformly distributed and 具g共R兲典 → 1 since particles with a very small Stokes number are distributed like fluid elements and particles with a very large Stokes number are not responsive to flow structures. For the intermediate Stokes numbers, particles interact with a range of fluid eddies and tend to accumulate in the regions of low vorticities, leading to the large values of 具g共R兲典 with a peak value around StK = 1 in the DNS. This trend is consistent with the previous observations.18,33 Our results from Figs. 8共a兲 and 9 are qualitatively similar to Fig. 22 of Fede and Simonin,14 where a different measure of preferential concentration was used. For very small Stokes numbers, StK ⬍ 1, the effect of filtering is to decrease the particle accumulation as small-scale eddies most relevant to accumulation have been removed. A very interesting observation from Fig. 9 is that the magnitude of the relative error in 具g共R兲典 for the FDNS flow has its maximum near StK = 0.5, with a maximum reduction of as much as 40%. For StK ⬎ 0.5, the magnitude of the relative error decreases with increasing the Stokes number, indicating that the SGS eddies make less and less contribution to the preferential concentration. For StK ⬎ 1, the FDNS yields a slightly higher 具g共R兲典 than the DNS result, indicating that the accumulation of these particles is more controlled by resolved eddies and the SGS eddies may act to randomize the particle distribution. Therefore, filtering out the random small eddies increases 具g共R兲典. For StK ⬎ 10, the particles are not responsive to the SGS eddies so the FDNS yields accurate results. Compared with the FDNS at the same spatial resolution, the LES predicts an even lower value of 具g共R兲典 for small Stokes numbers StK ⬍ 1. This is due to the overdissipation of the SGS model which tends to damp the eddies near the

101

102

SGS, making them contribute less to preferential concentration. Interestingly, for the intermediate Stokes numbers 1 ⬍ StK ⬍ 10, a higher 具g共R兲典 is realized when compared to the FDNS. This results from the fact that the resolved large-scale eddies are more coherent in space and time, particles have a longer interaction time with low vorticity regions, leading to a somewhat stronger preferential concentration. For the particles with very large Stokes numbers, StK ⬎ 10, the results from the DNS, FDNS, and LES are similar. The peak values of 具g共R兲典 in the FDNS and LES occur at a slightly larger Stokes numbers than that in the DNS. This is expected as the effective time scales for strongest vortical structures in the FDNS and LES are larger than K. 2. The radial relative velocity

For particles with large Stokes numbers, the velocities of the collision pair are uncorrelated and the relative velocity at contact is mainly determined by the energy-contained eddies.18 The energy-contained eddies are supposed to be at large scales and well resolved by LES. However, in the limit when particle inertia goes to zero, the relative velocity of particles is essentially determined by the local gradient of fluid velocity field provided that the collision radius R is small. In this limit, Saffman and Turner35 provided a prediction for 具兩wr共R兲兩典 in terms of the turbulent kinetic energy dissipation rate or the enstrophy of the flow field. As shown in Table II, although the relative percentage of turbulent kinetic energy that is filtered out in LES is negligible, a significant amount of enstrophy has been filtered out. Thus, we expect that a large error could arise in the LES prediction of the relative velocities for particles at very small Stokes numbers. Figure 10共a兲 shows the dependence of 具兩wr共R兲兩典 on the Stokes number, and Fig. 10共b兲 plots the relative errors of the FDNS and LES to the DNS results. It can be seen that 具兩wr共R兲兩典 is small when the particle Stokes number is small 共StK ⬍ 1兲. The reason for this behavior is that, in this limit, the two particles are strongly correlated as they are transported by large eddies, and only small-scale eddies are effective in generating particle relative motion. It is also observed that the quantity 具兩wr共R兲兩典 increases rapidly with the particle Stokes number for StK ⬍ 20. As the particle Stokes number increases, the large-scale turbulent fluctuations become effective in generating relative motion as different eddies encountered by the particles during their passage to a geometric

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055106-10

Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang

101 DNS

0

FDNS

B

ST, FDNS

C

ST, LES

0.5

A

A

A

B C

B C

B C

0.4A

A

0.3 B C

B C

0.2

B C

0.1 0

A

0.1

B C

0.2

0.3

0.4

LES ST, FDNS

B

-0.2

ST, LES

C B

B

B -0.15

C

-0.2

C

-0.25

-0.4

B

B

B

B

-0.35 C

C

C

C

-0.3

-0.4

0.5

StK

10

-2

10

-1

(a)

10 StK

0

10

1

10

-0.6

2

10

-2

10

-1

10 StK

(b)

0

contact can all contribute to the relative motion 共i.e., the memory effect兲. For StK ⬎ 20, the relative velocity begins to decrease slowly with the Stokes number, as there are no more large-scale fluctuations in the flow to disturb the particle motion.18 One can observe that the relative velocity in the FDNS is smaller than that in the DNS, but larger than that in the LES. Therefore, the relative error in the LES is larger than that in the FDNS for particles with small and intermediate Stokes numbers. As particle inertia increases, the radial relative velocity is affected more and more by the large-scale turbulence. The relative errors for FDNS and LES decrease with increasing particle Stokes number. We shall now analyze the various behaviors of the radial relative velocity with the existing theories. The radial relative velocity for finite Stokes number particles is governed by the interaction of particles with both large-scale energetic eddies and small-scale dissipation eddies. By assuming that both PDFs of wr and fluid velocity gradient u / x are Gaussian, Wang et al.18 developed a model for 具兩wr共R兲兩典 which combines the turbulent acceleration mechanism of the largescale turbulent motions and shear mechanism of the local small-scale dissipative motions as 具兩wr共R兲兩典 =

冑

2 2 具w 共R兲典 = r

冑

2 2 2 冑具wr,shear 典 + 具wr,accel 典. 共20兲

The acceleration mechanism36 was expressed as

冋

册 册

2 典 具wr,accel 2␥ 共1 + 2兲1/2 1 − = C w ␥−1 u ⬘2 1+

冋

0

0.1

0.2

0.3

0.4

0.5

StK

-1

10

FIG. 10. 共Color online兲 Effects of filtering operation and spectral eddy viscosity model on 具兩wr共R兲兩典 for particles at different Stokes numbers in LES 共643兲, FDNS 共kcf = 21兲, and DNS 共2563兲 flow fields. = 1.8 p / TE, Cw = 1.5 are used to fit the numerical results in Eq. 共21兲. ST in the legends denotes the ST theoretical prediction at zero Stokes number. 共a兲 具兩wr共R兲兩典 variation with Stk; 共b兲 relative error in 具兩wr共R兲兩典 of the LES and FDNS results using the DNS results as the benchmark.

FDNS

rErr of

A

rErr of

10

0

Eq.(20) ST, DNS

LES

1 1 ⫻ , 2 − 共1 + 兲 共1 + ␥兲2 / vK2 ,

1

10

2

2 具wr,shear 典

vK2

=

冉冊

1 R 15

2

共22兲

,

where R is the collision radius and was set to 0.5 in the simulation. Thus, the shear contribution scales with the small-scale flow velocity vK. The predictions of 具兩wr兩典 using Eq. 共20兲 are compared with the DNS results in Fig. 10共a兲. It is shown that Eq. 共20兲 can provide a reasonable prediction of the simulated 具兩wr兩典 data. It is noted that Eq. 共20兲 reduces to the Saffman and Turner’s 共ST兲 theory for particle with zero inertia or → 0, as 具兩wr共R兲兩典 =

冑 冑 2 R 15

,

共23兲

then 具兩wr共R兲兩典 is directly proportional to the square root of the dissipation rate in DNS, FDNS, and LES flow fields when R and are prescribed. The ST predictions of 具兩wr共R兲兩典 are 0.387, 0.285, and 0.254 for DNS, FDNS, and LES flow fields, respectively, and these are shown in Fig. 10共a兲 as well. The molecular viscosity coefficient used for the LES case is the same as the DNS case, = 0.0488. Indeed, for particles with very small Stokes numbers, the data agree with the ST predictions. This is true even for FDNS and LES when the actual dissipation rates 共see Table II兲 and molecular viscosity are used in the ST theory. The inset in Fig. 10共a兲 is an enlargement for the small StK region using linear scales, where additional data corresponding to fluid particles are added. The average radial relative velocity for fluid particles was calculated from the Eulerian velocity field using 具兩wr共R兲兩典 = 31 关具兩u共x,y,z,t兲 − u共x + R,y,z,t兲兩典 + 具兩v共x,y,z,t兲 − v共x,y + R,z,t兲兩典

共21兲

TE is the Eulerian intewhere = 2.5 p / TE, ␥ = 0.183u⬘ gral time scale, vK is the Kolmogorov velocity, and the factor Cw is used to fit the numerical results. Equation 共21兲 shows that the effect of acceleration mechanism on radial relative velocity depends on particle relative inertia , turbulent intensity u⬘2, and the turbulent Reynolds number Re since u⬘2 / vK2 = Re / 冑15. For particles with very small Stokes numbers, the relative velocity is related to the local shear rate of the flow35 2

10

+ 具兩w共x,y,z,t兲 − w共x,y,z + R,t兲兩典兴,

共24兲

where u, v, and w are the velocity along the x, y, and z axes, respectively. The numerical results of 具兩wr共R兲兩典 between fluid particles are 0.37, 0.281, and 0.238 for DNS, FDNS, and LES flow fields, respectively. Due to the loss of large amount of enstrophy in FDNS and LES flow fields, FDNS and LES both underpredict the relative velocities when compared to DNS for StK = 0. Furthermore, the relative reductions in FDNS and LES in this limit are different due to different levels of loss of enstrophy. The very minor discrepancies between our numerical results and the ST theory might come

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055106-11

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision 0.2 DNS FDNS LES

0

.

.

250 200

102

A

A

A

B C

B C

B C

150 AA

A

A

100 CBCB

B C

B C

.

10

-2

10

-1

10 StK

B

B

B

C

C

C

0

.

-0.4

B B B C C C

B C

-0.5 0

0

冑 冑 2

-0.2

-0.4

0

0.05

0.1

0.15

-0.6

10

1

10

2

0

10

(b)

1 2 v , 3 p

0.02

0.04

0.06

StK

-2

10

-1

10 StK

0

from the finite particle diameter of 0.5 assumed in the simulations since the ST theory assumes R Ⰶ . At the high Stokes number limit, one can observe in Fig. 10 that 具兩wr共R兲兩典 from DNS, FDNS, and LES are in good agreement and the relative errors of FDNS and LES approach zero. Since the velocities of the two colliding particles in this limit are completely uncorrelated and can be characterized by a Maxwellian distribution, the relative velocity is also Maxwellian. According to the analysis based on the kinetic theory for the very high Stokes number limit 共Stk → ⬁兲, 具兩wr共R兲兩典 can be expressed as33,37 具兩wr共R兲兩典 =

-0.1

-0.3

50

StK

(a)

FIG. 11. 共Color online兲 Effects of filtering operation ˙ 典 for particles and spectral eddy viscosity model on 具N c with different Stokes numbers in LES 共643兲, FDNS 共kcf = 21兲, and DNS 共2563兲 flow fields. ST in the legends denotes the Saffman and Turner’s theory for small ˙ 典 variation with St ; 共b兲 Stokes number particles. 共a兲 具N c k ˙ relative error of 具Nc典 from the LES and FDNS to the DNS results.

-0.2 rErr of

ST, DNS ST, FDNS ST, LES

rErr of

A B C

103

FDNS LES ST, FDNS ST, LES

B C

共25兲

where v2p is the variance of particle velocity. Since 具兩wr共R兲兩典 is now related to the one-point statistics v2p, which is almost unaffected by the small-scale fluid motion for particles with high Stokes numbers.14,38 Thus, the agreement of FDNS and LES with DNS at the high Stokes number limit shown in Fig. 10 are qualitatively consistent with this theoretical argument. A quantitative comparison in this limit will be provided in Sec. III E 3 when results on the collision rate are discussed. 3. Particle collision rate

Figure 11 shows the dynamically detected collision rate of particles from the DNS, FDNS, and LES. Again, the ST predictions are plotted for the small inertia limit using the dissipation rates in DNS, FDNS, and LES, respectively. The general trends from the three flow fields shown in Fig. 11共a兲 are similar. The collision rate increases very rapidly for small Stokes numbers, reaches a peak at StK = 10, and then drops slowly with an increase in StK. This behavior is consistent with the results for the collision kernel found by Sundaram and Collins33 and Zhou et al.39 The peak reflects both the effects of particle interaction with small and large scales of fluid motions. The interaction of heavy particle with smallscale motions produces a nonuniform particle distribution which enhances the collision rate through a large value of the radial distribution function at contact 具g共R兲典. On the other hand, as Stokes number increases, the turbulent transport effect which produces a large value of relative velocity, 具兩wr共R兲兩典, will dominate the collision rates.18 For small Stokes numbers, filtering causes a lower 具g共R兲典 and 具兩wr共R兲兩典, as shown in Figs. 9 and 10, and thus a lower par-

10

1

10

2

ticle collision rate according to Eq. 共18兲. The maximum relative errors occur at StK = 0.5 and are larger than 60%. For intermediate Stokes numbers, 1 ⬍ StK ⬍ 10, however, filtering causes an overestimation of 具g共R兲典 but an underestimation of 具兩wr共R兲兩典. This is a fortunate combination and leads to a rather accurate prediction of the overall collision rate. In fact, for StK ⬎ 3, both FDNS and LES yield very reasonable results for collision rate, and the difference between FDNS and LES is not noticeable. The nonmonotonic variation of the ˙ 典 for intermediate Stokes numbers relative errors in 具N c shown in Fig. 11共b兲 is a combination of the effects of 具g共R兲典 and 具兩wr共R兲兩典. In the two limiting cases of very small and very high Stokes numbers, particles uniformly distribute in the flow field, 具g共R兲典 ⬇ 1.0, the collision rate 具N˙c典 depends only on 具兩wr共R兲兩典. Based on Eqs. 共18兲 and 共23兲, the average collision rate for fluid particles 共the zero inertia limit兲 is 具N˙c典 =

冑

8 3 n20 R 15 2

冑

,

共26兲

where the average number density used in the simulations was n0 = 400 000/ 共2兲3 ⬇ 1613. Figure 11 shows that both FDNS and LES underpredict the collision rates due to the losses of dissipation rate in FDNS and LES flow fields, as implied by Eq. 共26兲. Based on Eqs. 共18兲 and 共25兲, the average collision rate for particles with very high Stokes numbers 共Stk → ⬁兲 is 具N˙c典 = 2冑n20R2冑 31 v2p .

共27兲

The quantitative comparison between the numerical collision rates and the theoretical prediction using Eq. 共27兲 for particles with high Stokes numbers is shown in Fig. 12, where ˜ = 具N˙ 典 / 共冑v2 / 3R2n2 / 2兲, the collision rate is normalized by N c

c

p

0

v2p is obtained from the simulated particle velocities in DNS, FDNS, and LES flow fields, respectively. The theoretical ˜ for high Stokes number limit is N ˜ = 4冑 = 7.09 value of N c c based on Eq. 共27兲. From Fig. 12, we observe that there is a good agreement in the collision rates obtained from DNS, FDNS, and LES. However, the theoretical value at the high Stokes number limit 共StK → ⬁兲 is still significantly above the values seen in the simulations. We note that StK = 80 for the DNS flow case is equivalent to p / TE ⬇ 5.7, where TE is the Eulerian integral time. Therefore, the theoretical limit would require a much larger p than those shown in Fig. 12. The

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055106-12

Phys. Fluids 22, 055106 共2010兲

Jin, He, and Wang

8

Ñc

6 4 DNS FDNS LES Abrahamson(1975)

2 0

40

60

80

StK FIG. 12. The comparison between the normalized numerical collision rates with the kinetic theory of Abrahamson 共Ref. 37兲 for particles with high Stokes numbers.

and LES with the spectral eddy viscosity SGS model tend to underpredict the radial distribution function at contact and radial relative velocity at contact. As a result, the error on the turbulent collision rate can become quite larger. For example, at StK = 0.5, the relative error on the collision rate can be as large as 60% in LES. By examining the spectral content of the particle concentration field in DNS, we have demonstrated that the dominant scales governing the preferential concentration change with the Stokes number. Those observations further clarify the dynamics of heavy particles relevant to the SGS motions in LES. Our analysis suggests that, for StK ⬍ 3, a particle SGS model must include the effects of the SGS motions on the turbulent collision of heavy particles.

ACKNOWLEDGMENTS

difficulty to reproduce the kinetic high Stokes number limit in DNS was noted in previous studies 共for example, see Fig. 4 in Ref. 33 and Fig. 6 in Ref. 40兲. IV. CONCLUSIONS

The turbulent collision of heavy particles is a small-scale process, where the small-scale velocity field plays an important role. This represents a challenge to LES of such a process. In LES, the large-scale velocity is explicitly obtained by solving the filtered Navier–Stokes equations which are closed with an eddy viscosity SGS model such as Eq. 共6兲 to account for the effects of the SGS velocity on the large-scale one, while the SGS velocity itself cannot be obtained explicitly. The particle-pair and collision-related statistics of heavy particles are significantly affected by the SGS velocity fields. We have demonstrated here that the errors in LES of turbulent collision can be attributed to two sources: the absence of SGS velocity due to filtering and the error from the eddy viscosity SGS model. We have attempted to clearly quantify the errors from these two sources using both a priori and a posteriori tests. We carefully examine the relative errors on the radial distribution functions at contact and the radial relative velocities at contact caused by filtering and the SGS model. The relative errors of collision-related statistics increase when the filter width increases, and for particles with small Stokes numbers, the relative errors are much larger than the relative amount of turbulent kinetic energy filtered out. For a given filter width, kcf = 21 or kcf = 0.284, these relative errors are found to be quite significant for a wide range of particle Stokes numbers, as a result of the interactions of particles with small-scale fluid motions. The errors exhibit nonmonotonic dependencies on the particle Stokes numbers. For a range of intermediate Stokes numbers, such as 1 ⬍ StK ⬍ 10, the radial distribution functions at contact in the FDNS and LES can be larger than that in the DNS. This nonmonotonic variations have a fortunate consequence: the collision rate can be well predicted by the FDNS and LES when StK ⬎ 3 since the overestimation of 具g共R兲典 partly cancels the underestimation of 具兩wr共R兲兩典. For StK ⬍ 1, we find that both FDNS

This work was supported by CAS 共Grant No. KJCX2SW-L08兲, 973 Program of China 共Grant No. 2007CB8 14800兲, NSFC 共Grant Nos. 10628206, 10732090, and 10702074兲, the LNM initial funding for young investigators, and SRF for ROCS, SEM. L.-P.W. acknowledges support by the U.S. National Science Foundation 共under Grant Nos. ATM-0527140 and ATM-0730766兲 during the course of this study and by NSFC 共Grant No. 10628206兲 which made it possible for him to visit and accomplish this work at LNM, Institute of Mechanics, Chinese Academy of Sciences in China. The constructive comments from two anonymous reviewers are greatly appreciated. 1

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055106-13

Phys. Fluids 22, 055106 共2010兲

Large-eddy simulation of turbulent collision

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